# Mean Rational Approximation for Compact Subsets with Thin Boundaries

John B. Conway1 and Liming Yang2 1Department of Mathematics, The George
Washington University, Washington, DC 20052<EMAIL_ADDRESS>2Department of
Mathematics, Virginia Polytechnic and State University, Blacksburg, VA 24061.
<EMAIL_ADDRESS>
###### Abstract.

In 1991, J. Thomson [13] obtained a celebrated decomposition theorem for
$P^{t}(\mu),$ the closed subspace of $L^{t}(\mu)$ spanned by the analytic
polynomials, when $1\leq t<\infty.$ Later, J. Brennan [3] generalized
Thomson’s theorem to $R^{t}(K,\mu),$ the closed subspace of $L^{t}(\mu)$
spanned by the rational functions with poles off a compact subset $K$
containing the support of $\mu,$ when the diameters of the components of
$\mathbb{C}\setminus K$ are bounded below. We extend the above decomposition
theorems for $R^{t}(K,\mu)$ when the boundary of $K$ is not too “wild”.

###### Key words and phrases:

Analytic Capacity, Analytic Bounded Point Evaluations, Bounded Point
Evaluations, and Mean Rational Approximation

###### 2010 Mathematics Subject Classification:

Primary 46E15; Secondary 30C85, 31A15, 47A15, 47B38

##  1\. Introduction

For a Borel subset $E$ of the complex plane $\mathbb{C},$ let $M_{0}(E)$
denote the set of finite complex-valued Borel measures that are compactly
supported in $E$ and let $M_{0}^{+}(E)$ be the set of positive measures in
$M_{0}(E).$ The support of $\nu\in M_{0}(\mathbb{C}),$ $\text{spt}(\nu),$ is
the smallest closed set that has full $|\nu|$ measure. For a compact subset
$K\subset\mathbb{C},$ $\mu\in M_{0}^{+}(K),$ and $1\leq t<\infty$, the
analytic polynomials and functions in $\mbox{Rat}(K):=\\{q:\mbox{$q$ is a
rational function with poles off $K$}\\}$ are members of $L^{t}(\mu)$. Let
$P^{t}(\mu)$ denote the closure of the (analytic) polynomials in $L^{t}(\mu)$
and let $R^{t}(K,\mu)$ denote the closure of $\mbox{Rat}(K)$ in $L^{t}(\mu)$.
Denote

$\ R^{t,\infty}(K,\mu)=R^{t}(K,\mu)\cap L^{\infty}(\mu).$

For $z_{0}\in\mathbb{C}$ and $r>0,$ let
$\mathbb{D}(z_{0},r):=\\{z:~{}|z-z_{0}|<r\\}.$ A point $z_{0}$ in $\mathbb{C}$
is called a bounded point evaluation for $P^{t}(\mu)$ (resp., $R^{t}(K,\mu)$)
if $f\mapsto f(z_{0})$ defines a bounded linear functional, denoted
$\phi_{z_{0}},$ for the analytic polynomials (resp., functions in
$\mbox{Rat}(K)$) with respect to the $L^{t}(\mu)$ norm. The collection of all
such points is denoted $\mbox{bpe}(P^{t}(\mu))$ (resp.,
$\mbox{bpe}(R^{t}(K,\mu)$)). If $z_{0}$ is in the interior of
$\mbox{bpe}(P^{t}(\mu))$ (resp., $\mbox{bpe}(R^{t}(K,\mu)$)) and there exist
positive constants $r$ and $M$ such that $|f(z)|\leq M\|f\|_{L^{t}(\mu)}$,
whenever $z\in\mathbb{D}(z_{0},r)$ and $f$ is an analytic polynomial (resp.,
$f\in\mbox{Rat}(K)$), then we say that $z_{0}$ is an analytic bounded point
evaluation for $P^{t}(\mu)$ (resp., $R^{t}(K,\mu)$). The collection of all
such points is denoted $\mbox{abpe}(P^{t}(\mu))$ (resp.,
$\mbox{abpe}(R^{t}(K,\mu))$). Actually, it follows from Thomson’s Theorem [13]
(or see Theorem 1.1, below) that $\mbox{abpe}(P^{t}(\mu))$ is the interior of
$\mbox{bpe}(P^{t}(\mu))$. This also holds in the context of $R^{t}(K,\mu)$ as
was shown by J. Conway and N. Elias in [7]. Now, $\mbox{abpe}(P^{t}(\mu))$ is
the largest open subset of $\mathbb{C}$ to which every function $f\in
P^{t}(\mu)$ has an analytic continuation $\rho(f)(z):=\phi_{z}(f)$ for
$z\in\mbox{abpe}(P^{t}(\mu)),$ that is,
$f(z)=\rho(f)(z)~{}\mu|_{\mbox{abpe}(P^{t}(\mu))}-a.a.,$ and similarly in the
context of $R^{t}(K,\mu)$. The map $\rho$ is called the evaluation map.

Let $S_{\mu}$ denote the multiplication by $z$ on $P^{t}(\mu)$ (resp.,
$R^{t}(K,\mu)$). The operator $S_{\mu}$ is pure if $P^{t}(\mu)$ (resp.,
$R^{t}(K,\mu)$) does not have a direct $L^{t}$ summand, and is irreducible if
$P^{t}(\mu)$ (resp., $R^{t}(K,\mu)$) contains no non-trivial characteristic
functions. For an open subset $U\subset\mathbb{C},$ we let $H^{\infty}(U)$
denote the bounded analytic functions on $U$. This is a Banach algebra in the
maximum modulus norm, but it is important to endow it also with a weak-star
topology.

We shall use $\overline{E}$ to denote the closure of the set
$E\subset\mathbb{C}$ and $\text{int}(E)$ to denote the interior of $E.$ Our
story begins with celebrated results of J. Thomson, in [13].

######  Theorem 1.1.

(Thomson 1991) Let $\mu\in M_{0}^{+}(\mathbb{C})$ and suppose that $1\leq
t<\infty$. Then there is a Borel partition $\\{\Delta_{i}\\}_{i=0}^{\infty}$
of $\mbox{spt}(\mu)$ such that

$\
P^{t}(\mu)=L^{t}(\mu|_{\Delta_{0}})\oplus\bigoplus_{i=1}^{\infty}P^{t}(\mu|_{\Delta_{i}})$

and the following statements are true:

(a) If $i\geq 1$, then $S_{\mu|_{\Delta_{i}}}$ on $P^{t}(\mu|_{\Delta_{i}})$
is irreducible.

(b) If $i\geq 1$ and $U_{i}:=abpe(P^{t}(\mu|_{\Delta_{i}}))$, then $U_{i}$ is
a simply connected region and $\Delta_{i}\subset\overline{U_{i}}$.

(c) If $S_{\mu}$ is pure (that is, if $\Delta_{0}=\emptyset$) and
$U=\text{abpe}(P^{t}(\mu))$, then the evaluation map $\rho$ is an isometric
isomorphism and a weak-star homeomorphism from $P^{t}(\mu)\cap
L^{\infty}(\mu)$ onto $H^{\infty}(U)$.

The next result is due to Brennan [3], which extends Theorem 1.1.

######  Theorem 1.2.

(Brennan 2008) Let $\mu\in M_{0}^{+}(K)$ for a compact set $K$. Suppose that
$1\leq t<\infty$ and the diameters of the components of $\mathbb{C}\setminus
K$ are bounded below. Then there is a Borel partition
$\\{\Delta_{i}\\}_{i=0}^{\infty}$ of $\mbox{spt}(\mu)$ and compact subsets
$\\{K_{i}\\}_{i=1}^{\infty}$ such that $\Delta_{i}\subset K_{i}$ for $i\geq
1$,

$\
R^{t}(K,\mu)=L^{t}(\mu|_{\Delta_{0}})\oplus\bigoplus_{i=1}^{\infty}R^{t}(K_{i},\mu|_{\Delta_{i}}),$

and the following statements are true:

(a) If $i\geq 1$, then $S_{\mu|_{\Delta_{i}}}$ on
$R^{t}(K_{i},\mu|_{\Delta_{i}})$ is irreducible.

(b) If $i\geq 1$ and $U_{i}:=\mbox{abpe}(R^{t}(K_{i},\mu|_{\Delta_{i}}))$,
then $U_{i}$ is connected and $K_{i}=\overline{U_{i}}$.

(c) If $i\geq 1$, then the evaluation map $\rho_{i}$ is an isometric
isomorphism and a weak-star homeomorphism from
$R^{t,\infty}(K_{i},\mu|_{\Delta_{i}})$ onto $H^{\infty}(U_{i})$.

If $B\subset\mathbb{C}$ is a compact subset, then we define the analytic
capacity of $B$ by

$\ \gamma(B)=\sup|f^{\prime}(\infty)|,$

where the supremum is taken over all those functions $f$ that are analytic in
$\mathbb{C}_{\infty}\setminus B$
($\mathbb{C}_{\infty}:=\mathbb{C}\cup\\{\infty\\}$) such that $|f(z)|\leq 1$
for all $z\in\mathbb{C}_{\infty}\setminus B$; and
$f^{\prime}(\infty):=\lim_{z\rightarrow\infty}z[f(z)-f(\infty)].$ The analytic
capacity of a general subset $E$ of $\mathbb{C}$ is given by:
$\gamma(E)=\sup\\{\gamma(B):B\subset E\text{ compact}\\}.$ We will write
$\gamma-a.a.$, for a property that holds everywhere, except possibly on a set
of analytic capacity zero. Good sources for basic information about analytic
capacity are Chapter VIII of [10], Chapter V of [6], and [16].

We let $\partial_{e}K$ (the exterior boundary of $K$) denote the union of the
boundaries of all the components of $\mathbb{C}\setminus K.$ Define

$\displaystyle\ \partial_{1}K=\left\\{\lambda\in
K:~{}\underset{\delta\rightarrow
0}{\overline{\lim}}\dfrac{\gamma(\mathbb{D}(\lambda,\delta)\setminus
K)}{\delta}>0\right\\}.$ (1.1)

Obviously, $\partial_{e}K\subset\partial_{1}K\subset\partial K.$ If the
diameters of the components of $\mathbb{C}\setminus K$ are bounded below, then
there exist $\epsilon_{0}>0$ and $\delta_{0}>0$ such that for each
$\lambda\in\partial K,$

$\displaystyle\ \gamma(\mathbb{D}(\lambda,\delta)\setminus
K)\geq\epsilon_{0}\delta\text{ for }\delta<\delta_{0}.$ (1.2)

Clearly, if $K$ satisfies (1.2), then $\partial K=\partial_{1}K.$ Conversely,
it is straightforward to construct a compact subset $K$ such that $\partial
K=\partial_{1}K$ and $K$ does not satisfy (1.2).

In [8, Section 5], Conway and Yang introduced a concept of non-removable
boundary for $R^{t}(K,\mu)$ to identify $R^{t,\infty}(K,\mu)$ with a sub-
algebra of $H^{\infty}(\text{int}(K))$ when $K$ is a string of beads set (see
[8, Main Theorem]). In this paper, we extend the concept of non-removable
boundary, denoted $\mathcal{F},$ to an arbitrary compact subset $K$ and
$\mu\in M_{0}^{+}(K)$ (see Definition 3.4 and Definition 5.1). Intuitively,
the non-removable boundary $\mathcal{F}$ splits into three sets,
$\mathcal{F}_{0}$ and $\mathcal{F}_{+}\cup\mathcal{F}_{-}$ such that (1)
Cauchy transforms of annihilating measures $g\mu$ ($g\perp R^{t}(K,\mu)$) are
zero on $\mathcal{F}_{0}$ and (2) Cauchy transforms of annihilating measures
$g\mu$ have zero “one side non-tangential limits” on
$\mathcal{F}_{+}\cup\mathcal{F}_{-}.$ We show in Proposition 5.3 that

$\displaystyle\ \partial_{1}K\subset\mathcal{F},~{}\gamma-a.a..$ (1.3)

The aim of this paper is to prove the following main theorem.

######  Theorem A.

Let $K\subset\mathbb{C}$ be a compact subset, $1\leq t<\infty,$ and $\mu\in
M_{0}^{+}(K).$ Let $\text{abpe}(R^{t}(K,\mu))=\cup_{i=1}^{\infty}U_{i},$ where
$U_{i}$ is a connected component. If $\partial U_{i}\cap\partial
K\subset\mathcal{F},~{}\gamma-a.a.$ for all $i\geq 1,$ then there is a Borel
partition $\\{\Delta_{i}\\}_{i=0}^{\infty}$ of $\mbox{spt}(\mu)$ and compact
subsets $\\{K_{i}\\}_{i=0}^{\infty}$ such that $\Delta_{i}\subset K_{i}$ for
$i\geq 0$,

$\
R^{t}(K,\mu)=R^{t}(K_{0},\mu|_{\Delta_{0}})\oplus\bigoplus_{i=1}^{\infty}R^{t}(K_{i},\mu|_{\Delta_{i}}),$

and the following properties hold:

(a) $K_{0}$ is the spectrum of $S_{\mu|_{\Delta_{0}}}$ and
$\text{abpe}(R^{t}(K_{0},\mu|_{\Delta_{0}}))=\emptyset.$

(b) If $i\geq 1$, then $S_{\mu|_{\Delta_{i}}}$ on
$R^{t}(K_{i},\mu|_{\Delta_{i}})$ is irreducible.

(c) If $i\geq 1,$ then $U_{i}=\mbox{abpe}(R^{t}(K_{i},\mu|_{\Delta_{i}}))$ and
$K_{i}=\overline{U_{i}}$.

(d) If $i\geq 1$, then the evaluation map $\rho_{i}:f\rightarrow f|_{U_{i}}$
is an isometric isomorphism and a weak-star homeomorphism from
$R^{t,\infty}(K_{i},\mu|_{\Delta_{i}})$ onto $H^{\infty}(U_{i})$.

Consequently, by (1.3), we prove the following theorem in section 6, which
extends both Theorem 1.1 and Theorem 1.2.

######  Theorem B.

Let $K$ be a compact set such that $\gamma(\partial
K\setminus\partial_{1}K)=0$. Suppose that $1\leq t<\infty$ and $\mu\in
M_{0}^{+}(K).$ Then there is a Borel partition
$\\{\Delta_{i}\\}_{i=0}^{\infty}$ of $\mbox{spt}(\mu)$ and compact subsets
$\\{K_{i}\\}_{i=1}^{\infty}$ such that $\Delta_{i}\subset K_{i}$ for $i\geq
1$,

$\
R^{t}(K,\mu)=L^{t}(\mu|_{\Delta_{0}})\oplus\bigoplus_{i=1}^{\infty}R^{t}(K_{i},\mu|_{\Delta_{i}}),$

and the following statements are true:

(a) If $i\geq 1$, then $S_{\mu|_{\Delta_{i}}}$ on
$R^{t}(K_{i},\mu|_{\Delta_{i}})$ is irreducible.

(b) If $i\geq 1$ and $U_{i}:=\mbox{abpe}(R^{t}(K_{i},\mu|_{\Delta_{i}}))$,
then $U_{i}$ is connected and $K_{i}=\overline{U_{i}}$.

(c) If $i\geq 1$, then the evaluation map $\rho_{i}:f\rightarrow f|_{U_{i}}$
is an isometric isomorphism and a weak-star homeomorphism from
$R^{t,\infty}(K_{i},\mu|_{\Delta_{i}})$ onto $H^{\infty}(U_{i})$.

Main Theorem in [8] provides an interesting example of $R^{t}(K,\mu)$ that
does not satisfy the assumptions of Theorem A (also see [8, Proposition 3.4]).
In section 2, we review some results of analytic capacity and Cauchy transform
that are needed in our analysis. Theorem 2.11 proves a Plemelj formula for the
Cauchy transform of an arbitrary finite complex-values Borel measure. In
section 3, we introduce the concepts of non-removable boundary and removable
set for an arbitrary compact subset $K\subset\mathbb{C}$ and $\mu\in
M_{0}^{+}(K)$ and prove Theorem 3.7 to characterize them. [8] provides a good
example of the concepts for a string of beads set. Section 4 proves a key
lemma (Lemma 4.1) for our approximation scheme. We discuss the relationship
between the removable set and $\text{abpe}(R^{t}(K,\mu))$ in section 5
(Theorem 5.5). In section 6, we first review the modified Vitushkin
approximation scheme of Paramonov (Lemma 6.7). Then, we prove Theorem A and
Theorem B.

##  2\. Preliminaries

For $\nu\in M_{0}(\mathbb{C})$ and $\epsilon>0,$ $\mathcal{C}_{\epsilon}(\nu)$
is defined by

$\ \mathcal{C}_{\epsilon}(\nu)(z)=\int_{|w-z|>\epsilon}\dfrac{1}{w-z}d\nu(w).$

The (principal value) Cauchy transform of $\nu$ is defined by

$\displaystyle\ \mathcal{C}(\nu)(z)=\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(\nu)(z)\ $ (2.1)

for all $z\in\mathbb{C}$ for which the limit exists. From Corollary 2.3 below,
we see that (2.1) is defined for all $z$ except for a set of zero analytic
capacity. In particular, it is defined for almost all $z$ with respect to the
area measure (the Lebesgue measure on $\mathbb{C}$) ${\mathfrak{m}}.$
Throughout this paper, the Cauchy transform of a measure always means the
principal value of the transform. In the sense of distributions,

$\displaystyle\ \bar{\partial}\mathcal{C}(\nu)=-\pi\nu.$ (2.2)

The maximal Cauchy transform is defined by

$\ \mathcal{C}_{*}(\nu)(z)=\sup_{\epsilon>0}|\mathcal{C}_{\epsilon}(\nu)(z)|.$

A related capacity, $\gamma_{+},$ is defined for subsets $E$ of $\mathbb{C}$
by:

$\ \gamma_{+}(E)=\sup\|\mu\|,$

where the supremum is taken over $\mu\in M_{0}^{+}(E)$ for which
$\|\mathcal{C}(\mu)\|_{L^{\infty}(\mathbb{C})}\leq 1.$ Since $\mathcal{C}\mu$
is analytic in $\mathbb{C}_{\infty}\setminus\mbox{spt}(\mu)$ and
$|(\mathcal{C}(\mu)^{\prime}(\infty)|=\|\mu\|$, we have:
$\gamma_{+}(E)\leq\gamma(E)$.

X. Tolsa has established the following astounding results. See [15] (also
Theorem 6.1 and Corollary 6.3 in [16]) for (1) and (2). See [14, Proposition
2.1] (also [16, Proposition 4.16]) for (3).

######  Theorem 2.1.

(Tolsa 2003) (1) $\gamma_{+}$ and $\gamma$ are actually equivalent. That is,
there is an absolute constant $A_{T}$ such that for all $E\subset\mathbb{C},$

$\ \gamma(E)\leq A_{T}\gamma_{+}(E).$

(2) Semiadditivity of analytic capacity: for
$E_{1},E_{2},...,E_{m}\subset\mathbb{C}$ ($m$ may be $\infty$),

$\ \gamma\left(\bigcup_{i=1}^{m}E_{i}\right)\leq
A_{T}\sum_{i=1}^{m}\gamma(E_{i}).$

(3) There is an absolute positive constant $C_{T}$ such that for $a>0$, we
have:

$\ \gamma(\\{\mathcal{C}_{*}(\nu)\geq a\\})\leq\dfrac{C_{T}}{a}\|\nu\|.$

For $\eta\in M_{0}^{+}(\mathbb{C})$, define

$\
N_{2}(\eta)=\sup_{\epsilon>0}\sup_{\|f\|_{L^{2}(\eta)}=1}\|\mathcal{C}_{\epsilon}(f\eta)\|_{L^{2}(\eta)}.$

$\eta$ is $c$-linear growth if $\eta(\mathbb{D}(\lambda,\delta))\leq
c\delta,\text{ for }\lambda\in\mathbb{C}\text{ and }\delta>0.$ We use
$C_{1},C_{2},...$ for absolute constants that may change from one step to the
next. The following Proposition is from [16, Theorem 4.14] and its proofs.

######  Proposition 2.2.

If $F\subset\mathbb{C}$ is a compact subset and $\eta\in M_{0}(F),$ then the
following properties are true.
(1) If $\|{\mathcal{C}}\eta\|_{L^{\infty}(\mathbb{C})}\leq 1,$ then $\eta$ is
$1$-linear growth and
$\sup_{\epsilon>0}\|\mathcal{C}_{\epsilon}(\eta)\|_{L^{\infty}(\mathbb{C})}\leq
C_{1}.$
(2) If $\eta$ is $1$-linear growth and
$\|\mathcal{C}_{\epsilon}(\eta)\|_{L^{\infty}(\mathbb{C})}\leq 1$ for all
$\epsilon>0,$ then there exists a subset $A\subset F$ such that $\eta(F)\leq
2\eta(A)$ and $N_{2}(\eta|_{A})\leq C_{2}.$
(3) If $N_{2}(\eta)\leq 1,$ then there exists some function $w$ supported on
$F$, with $0\leq w\leq 1$ such that $\eta(F)\ \leq\ 2\int wd\eta$ and
$\sup_{\epsilon>0}\|\mathcal{C}_{\epsilon}(w\eta)\|_{L^{\infty}(\mathbb{C})}\
\leq\ C_{3}.$

Combining Theorem 2.1 (1), Proposition 2.2, and [15] (or [16, Theorem 8.1]),
we get the following corollary. The reader may also see [1, Corollary 3.1].

######  Corollary 2.3.

If $\nu\in M_{0}(\mathbb{C}),$ then there exists
$\mathcal{Q}\subset\mathbb{C}$ with $\gamma(\mathcal{Q})=0$ such that
$\lim_{\epsilon\rightarrow 0}\mathcal{C}_{\epsilon}(\nu)(z)$ exists for
$z\in\mathbb{C}\setminus\mathcal{Q}$.

######  Corollary 2.4.

Let $E_{n}\subset E_{n+1}\subset\mathbb{D}(0,R)$ be a sequence of subsets.
Then

$\ \gamma\left(\cup_{n=1}^{\infty}E_{n}\right)\leq
C_{1}\lim_{n\rightarrow\infty}\gamma(E_{n}).$

###### Proof.

By Theorem 2.1 (1) and Proposition 2.2 (2), there exists a compact subset
$F\subset\cup_{n=1}^{\infty}E_{n}$ and $\eta\in M_{0}^{+}(F)$ with $1$-linear
growth such that $N_{2}(\eta)=1$ and

$\ \gamma\left(\cup_{n=1}^{\infty}E_{n}\right)\leq C_{4}\|\eta\|\leq
C_{4}\lim_{n\rightarrow\infty}\eta(E_{n}).$

Since $N_{2}(\eta|_{E_{n}})\leq 1,$ by Proposition 2.2 (3) and Theorem 2.1
(1), we have $\eta(E_{n})\leq C_{5}\gamma(E_{n}).$ ∎

######  Lemma 2.5.

Let $\\{\nu_{j}\\}\subset M_{0}(\mathbb{C}).$ Then for $\epsilon>0$, there
exists a Borel subset $F$ such that $\gamma(F^{c})<\epsilon$ and
$\mathcal{C}_{*}(\nu_{j})(z)\leq M_{j}<\infty$ for $z\in F$.

###### Proof.

Let $A_{j}=\\{\mathcal{C}_{*}(\nu_{j})(z)\leq M_{j}\\}.$ By Theorem 2.1 (3),
we can select $M_{j}>0$ so that
$\gamma(A_{j}^{c})<\frac{\epsilon}{2^{j+1}A_{T}}.$ Set
$F=\cap_{j=1}^{\infty}A_{j}$. Then applying Theorem 2.1 (2), we get

$\ \gamma(F^{c})\leq A_{T}\sum_{j=1}^{\infty}\gamma(A_{j}^{c})<\epsilon.$

∎

Given $0<\epsilon<\infty$, for $A\subset\mathbb{C}$, we define

$\
\mathcal{H}^{1}_{\epsilon}(A)=\inf\left\\{\sum_{i}\text{diam}(A_{i}):~{}A\subset\cup_{i}A_{i},~{}\text{diam}(A_{i})\leq\epsilon\right\\}.$

The $1$-dimensional Hausdorff measure of $A$ is:

$\
\mathcal{H}^{1}(A)=\sup_{\epsilon>0}\mathcal{H}^{1}_{\epsilon}(A)=\lim_{\epsilon\rightarrow
0}\mathcal{H}^{1}_{\epsilon}(A).$

For detailed information about Hausdorff measures, the reader may take a look
at the following books: Carleson [4] and Mattila [11].

For $\nu\in M_{0}(\mathbb{C}),$ define the zero and non zero linear density
sets:

$\
\mathcal{ZD}(\nu)=\\{\lambda:~{}\Theta_{\nu}(\lambda)=0\\},~{}\mathcal{ND}(\nu)=\\{\lambda:~{}\Theta^{*}_{\nu}(\lambda)>0\\},$

where $\Theta_{\nu}(\lambda):=\lim_{\delta\rightarrow
0}\frac{|\nu|(\mathbb{D}(\lambda,\delta))}{\delta}$ if the limit exists and
$\Theta_{\nu}^{*}(\lambda):=\underset{\delta\rightarrow
0}{\overline{\lim}}\frac{|\nu|(\mathbb{D}(\lambda,\delta))}{\delta}.$ Set

$\mathcal{ND}(\nu,n)=\\{\lambda:~{}n^{-1}\leq\Theta^{*}_{\nu}(\lambda)\leq
n\\}.$

######  Lemma 2.6.

Let $\mu\in M_{0}^{+}(\mathbb{C})$ and $\nu\in M_{0}(\mathbb{C}).$ The
following properties hold.

(a) For $\lambda_{2}\geq\lambda_{1}>0$, let
$E=\\{z:~{}\Theta^{*}_{\mu}(z)\geq\lambda_{1}\\}$ and
$F=\\{z:~{}\lambda_{1}\leq\Theta^{*}_{\mu}(z)\leq\lambda_{2}\\}.$ Then
$\mathcal{H}^{1}(E)\leq C_{6}\frac{\mu(E)}{\lambda_{1}}$ and
$\mu|_{F}=g\mathcal{H}^{1}|_{F}$, where $g$ is some Borel function such that
$C_{7}\lambda_{1}\leq g(z)\leq C_{8}\lambda_{2},~{}\mathcal{H}^{1}|_{F}-a.a.$.

(b) If $\mathcal{Q}_{\mu}=\\{z:~{}\Theta_{\mu}^{*}(z)=\infty\\}$, then
$\gamma(\mathcal{Q}_{\mu})=\mathcal{H}^{1}(\mathcal{Q}_{\mu})=0$,

$\ \mathcal{ND}(\mu)=\bigcup_{n}\mathcal{ND}(\mu,n)\cup\mathcal{Q}_{\mu},$

$\mu|_{\mathcal{ND}(\mu)}=g\mathcal{H}^{1}|_{\mathcal{ND}(\mu)}$, and
$C_{7}\frac{1}{n}\leq g(z)\leq
C_{8}n,~{}\mathcal{H}^{1}|_{\mathcal{ND}(\mu,n)}-a.a.$.

(c) If $\eta\in M_{0}^{+}(\mathbb{C})$ is $c$-linear growth and
$\eta\perp|\nu|$, then $\eta(\mathcal{ND}(\nu))=0$.

###### Proof.

See [16, Lemma 8.12] for (a).

(b): From (a),
$\mathcal{H}^{1}(\mathcal{Q}_{\mu})\leq\mathcal{H}^{1}\\{z:~{}\Theta^{*}_{\mu}(z)\geq
n\\}\leq\frac{C_{4}}{n}\|\mu\|,$ which implies (b).

(c): By (a), $\mathcal{H}^{1}(\mathcal{ND}(\eta,m)\cap\mathcal{ND}(\nu,n))=0$
as $\eta\perp|\nu|$. Therefore,
$\mathcal{H}^{1}(\mathcal{ND}(\eta)\cap\mathcal{ND}(\nu))=0$. On the other
hand, $\mathcal{H}^{1}(\mathcal{ND}(\nu,n))<\infty$, which implies
$\eta|_{\mathcal{ND}(\nu)}$ is absolutely continuous with respect to
$\mathcal{H}^{1}|_{\mathcal{ND}(\nu)}$ since $\eta$ is linear growth. Hence,
$\eta(\mathcal{ZD}(\eta)\cap\mathcal{ND}(\nu))=0$, which implies
$\eta(\mathcal{ND}(\nu))=0$. ∎

######  Definition 2.7.

Let $\mathcal{Q}$ be a set with $\gamma(\mathcal{Q})=0$. Let $f(z)$ be a
function defined on $\mathbb{D}(\lambda,\delta_{0})\setminus\mathcal{Q}$ for
some $\delta_{0}>0.$ The function $f$ has a $\gamma$-limit $a$ at $\lambda$ if

$\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda,\delta)\cap\\{|f(z)-a|>\epsilon\\})}{\delta}=0$

for all $\epsilon>0$. If in addition, $f(\lambda)$ is well defined and
$a=f(\lambda)$, then $f$ is $\gamma$-continuous at $\lambda$.

The following lemma is from [1, Lemma 3.2].

######  Lemma 2.8.

Let $\nu\in M_{0}(\mathbb{C})$ and assume that for some
$\lambda_{0}\in\mathbb{C},$ $\Theta_{\nu}(\lambda_{0})=0$ and
$\mathcal{C}(\nu)(\lambda_{0})=\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(\nu)(\lambda_{0})$ exists. Then the Cauchy transform
$\mathcal{C}(\nu)(\lambda)$ is $\gamma$-continuous at $\lambda_{0}$.

Let $\mathbb{R}$ be the real line. Let $A:\mathbb{R}\rightarrow\mathbb{R}$ be
a Lipschitz function. The Lipschitz graph $\Gamma$ of $A$ is defined by
$\Gamma=\left\\{(x,A(x)):~{}x\in\mathbb{R}\right\\}$. We define, for
$\lambda\in\Gamma,$ the open upper cone

$\
UC(\lambda,\alpha)=\left\\{z\in\mathbb{C}:~{}|Re(z)-Re(\lambda)|<\alpha(Im(z)-Im(\lambda))\right\\}$

and the open lower cone

$\
LC(\lambda,\alpha)=\left\\{z\in\mathbb{C}:~{}|Re(z)-Re(\lambda)|<-\alpha(Im(z)-Im(\lambda))\right\\}.$

Set
$UC(\lambda,\alpha,\delta)=UC(\lambda,\alpha)\cap\mathbb{D}(\lambda,\delta)$
and
$LC(\lambda,\alpha,\delta)=LC(\lambda,\alpha)\cap\mathbb{D}(\lambda,\delta)$.

If $\alpha<\frac{1}{\|A^{\prime}\|_{\infty}},$ then, for almost all
$\lambda\in\Gamma,$ there exists $\delta>0$ such that

$\ UC(\lambda,\alpha,\delta)\subset
U_{\Gamma}:=\left\\{z\in\mathbb{C}:~{}Im(z)>A(Re(z))\right\\}$

and

$\ LC(\lambda,\alpha,\delta)\subset
L_{\Gamma}:=\left\\{z\in\mathbb{C}:~{}Im(z)<A(Re(z))\right\\}.$

We consider the usual complex-valued measure

$\ \dfrac{1}{2\pi i}dz_{\Gamma}=\dfrac{1+iA^{\prime}(Re(z))}{2\pi
i(1+A^{\prime}(Re(z))^{2})^{\frac{1}{2}}}d\mathcal{H}^{1}|_{\Gamma}=L(z)d\mathcal{H}^{1}|_{\Gamma}.$

Notice that $|L(z)|=\frac{1}{2\pi}.$ Plemelj’s formula for Lipschitz graphs as
in [16, Theorem 8.8] is the following.

######  Lemma 2.9.

Let $A:\mathbb{R}\rightarrow\mathbb{R}$ be a Lipschitz function with its graph
$\Gamma$. Let $0<\alpha<\frac{1}{\|A^{\prime}\|_{\infty}}$. Then, for all
$f\in L^{1}(\mathcal{H}^{1}|_{\Gamma})$, the non-tangential limits of
$\mathcal{C}(fdz_{\Gamma})(\lambda)$ exists for
$\mathcal{H}^{1}|_{\Gamma}-a.a.$ and the following identities hold
$\mathcal{H}^{1}|_{\Gamma}-a.a.$ for $\lambda\in\Gamma$:

$\ \dfrac{1}{2\pi i}\lim_{z\in
UC(\lambda,\alpha)\rightarrow\lambda}\mathcal{C}(fdz_{\Gamma})(z)=\dfrac{1}{2\pi
i}\mathcal{C}(fdz_{\Gamma})(\lambda)+\dfrac{1}{2}f(\lambda),$

and

$\ \dfrac{1}{2\pi i}\lim_{z\in
LC(\lambda,\alpha)\rightarrow\lambda}\mathcal{C}(fdz_{\Gamma})(z)=\dfrac{1}{2\pi
i}\mathcal{C}(fdz_{\Gamma})(\lambda)-\dfrac{1}{2}f(\lambda).$

######  Lemma 2.10.

Let $A:\mathbb{R}\rightarrow\mathbb{R}$ be a Lipschitz function with graph
$\Gamma$. Then $N_{2}(\mathcal{H}^{1}|_{\Gamma})<\infty,$ while
$N_{2}(\mathcal{H}^{1}|_{\Gamma})$ only depends on $\|A^{\prime}\|_{\infty}$.
Consequently, there is a constant $c_{\Gamma}>0$ that depends only on
$\|A^{\prime}\|_{\infty}$ such that for $E\subset\Gamma$,

$\displaystyle\
c_{\Gamma}\mathcal{H}^{1}|_{\Gamma}(E)\leq\gamma(E)\leq\mathcal{H}^{1}|_{\Gamma}(E).$
(2.3)

###### Proof.

See [5] or [16, Theorem 3.11] for $N_{2}(\mathcal{H}^{1}|_{\Gamma})<\infty.$
Therefore, from Proposition 2.2 (3) and Theorem 2.1 (1), we conclude that
$c_{\Gamma}\mathcal{H}^{1}|_{\Gamma}(E)\leq\gamma(E)$ holds. See [16, Theorem
1.21] for $\gamma(E)\leq\mathcal{H}^{1}|_{\Gamma}(E).$ ∎

We now improve Lemma 2.9 with the following theorem.

######  Theorem 2.11.

(Plemelj’s Formula for an arbitrary measure) Let
$\alpha<\frac{1}{\|A^{\prime}\|_{\infty}}$ and $\nu\in M_{0}(\mathbb{C})$.
Suppose that $\nu=h\mathcal{H}^{1}|_{\Gamma}+\nu_{s}$ is the Radon-Nikodym
decomposition with respect to $\mathcal{H}^{1}|_{\Gamma}$, where $h\in
L^{1}(\mathcal{H}^{1}|_{\Gamma})$ and $\nu_{s}\perp\mathcal{H}^{1}|_{\Gamma}$.
Then there exists a subset $\mathcal{Q}\subset\mathbb{C}$ with
$\gamma(\mathcal{Q})=0$, such that the following hold:

(a) $\mathcal{C}(\nu)(\lambda_{0})=\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(\nu)(\lambda_{0})$ exists for
$\lambda_{0}\in\mathbb{C}\setminus\mathcal{Q}$.

(b) For $\lambda_{0}\in\Gamma\setminus\mathcal{Q}$ and $\epsilon>0$,
$v^{+}(\nu,\Gamma)(\lambda_{0}):=\mathcal{C}(\nu)(\lambda_{0})+\frac{h(\lambda_{0})}{2L(\lambda_{0})},$

$\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(U_{\Gamma}\cap\mathbb{D}(\lambda,\delta)\cap\\{|\mathcal{C}(\nu)(\lambda)-v^{+}(\nu,\Gamma)(\lambda_{0})|>\epsilon\\})}{\delta}=0.$

(c) For $\lambda_{0}\in\Gamma\setminus\mathcal{Q}$ and $\epsilon>0$,
$v^{-}(\nu,\Gamma)(\lambda_{0}):=\mathcal{C}(\nu)(\lambda_{0})-\frac{h(\lambda_{0})}{2L(\lambda_{0})},$

$\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(L_{\Gamma}\cap\mathbb{D}(\lambda,\delta)\cap\\{|\mathcal{C}(\nu)(\lambda)-v^{-}(\nu,\Gamma)(\lambda_{0})|>\epsilon\\})}{\delta}=0.$

(d) For $\lambda_{0}\in\Gamma\setminus\mathcal{Q}$ and $\epsilon>0$,
$v^{0}(\nu,\Gamma)(\lambda_{0}):=\mathcal{C}(\nu)(\lambda_{0}),$

$\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(\Gamma\cap\mathbb{D}(\lambda,\delta)\cap\\{|\mathcal{C}(\nu)(\lambda)-v^{0}(\nu,\Gamma)(\lambda_{0})|>\epsilon\\})}{\delta}=0.$

As $\nu$ is compactly supported, We will just consider the portion of the
graph of $\Gamma$ that lies in $\\{z:~{}|Re(z)|<r\\}$ for some $r>0.$ Since
$\nu_{s}\perp\mathcal{H}^{1}|_{\Gamma}$, using Lemma 2.6 (c) and Lemma 2.8, we
conclude that (b), (c), and (d) holds for the measure $\nu_{s}$. So we may
assume that $\nu=hd\mathcal{H}^{1}|_{\Gamma}$.

###### Proof.

(Theorem 2.11 (d)): Since $\gamma|_{\Gamma}\approx\mathcal{H}^{1}|_{\Gamma}$
by (2.3), we use $\mathcal{H}^{1}|_{\Gamma}$ instead of $\gamma$ in the proof.

From Lemma 2.10, we see that the Cauchy transform of
$\mathcal{H}^{1}|_{\Gamma}$ is bounded on $L^{2}(\mathcal{H}^{1}|_{\Gamma})$.
So we get $\mathcal{C}\mathcal{H}^{1}|_{\Gamma}\in
L^{2}(\mathcal{H}^{1}|_{\Gamma})$ as $\Gamma\subset\\{z:~{}|Re(z)|<r\\}$. Also
$h\in L^{1}(\mathcal{H}^{1}|_{\Gamma})$. Let $\lambda_{0}$ be a Lebesgue point
of $h$ and $\mathcal{C}\mathcal{H}^{1}|_{\Gamma}$. That is,

$\displaystyle\ \lim_{\delta\rightarrow
0}\dfrac{1}{\delta}\int_{\mathbb{D}(\lambda_{0},\delta)}|h(z)-h(\lambda_{0})|d\mathcal{H}^{1}|_{\Gamma}(z)=0$
(2.4)

and

$\displaystyle\ \lim_{\delta\rightarrow
0}\dfrac{1}{\delta}\int_{\mathbb{D}(\lambda_{0},\delta)}|\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(z)-\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(\lambda_{0})|d\mathcal{H}^{1}|_{\Gamma}(z)=0$
(2.5)

as $\mathcal{H}^{1}|_{\Gamma}(\mathbb{D}(\lambda_{0},\delta))\approx\delta.$

For $\epsilon>0$, we get (assuming $h(\lambda_{0})\neq 0$):

$\ \begin{aligned} \
&\left\\{|\mathcal{C}\nu(\lambda)-\mathcal{C}\nu(\lambda_{0})|>\epsilon\right\\}\\\
\
\subset&\left\\{|\mathcal{C}((h-h(\lambda_{0}))\mathcal{H}^{1}|_{\Gamma})(\lambda)-\mathcal{C}((h-h(\lambda_{0}))\mathcal{H}^{1}|_{\Gamma})(\lambda_{0})|>\frac{\epsilon}{2}\right\\}\\\
\
&\bigcup\left\\{|\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(\lambda)-\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(\lambda_{0})|>\frac{\epsilon}{2|h(\lambda_{0})|}\right\\}.\
\end{aligned}$

It follows from (2.5) that

$\ \lim_{\delta\rightarrow
0}\dfrac{\mathcal{H}^{1}|_{\Gamma}\left(\left\\{|\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(\lambda)-\mathcal{C}\mathcal{H}^{1}|_{\Gamma}(\lambda_{0})|>\dfrac{\epsilon}{2|h(\lambda_{0})|}\right\\}\cap\mathbb{D}(\lambda_{0},\delta)\right)}{\delta}=0.$

Therefore, we may assume that $h(\lambda_{0})=0$. In this case, $\nu$
satisfies the assumptions of Lemma 2.8 as (2.4) holds and we get (d) because
$\mathcal{C}(\nu)(z)$ is $\gamma$-continuous at $\lambda_{0}$ by Lemma 2.8. ∎

For a complex-valued function $f$ defined on $\mathbb{C},$ recall that

$\ \mathcal{N}(f)=\\{\lambda:~{}f(\lambda)\neq
0\\},~{}\mathcal{Z}(f)=\\{\lambda:~{}f(\lambda)=0\\}$

###### Proof.

(Theorem 2.11 (b)): For $a>0$, let $A_{n}$ be the set of
$\lambda\in\mathcal{N}(h)$ satisfying

$\
\left|\mathcal{C}(\nu)(z)-\mathcal{C}(\nu)(\lambda)-\frac{h(\lambda)}{2L(\lambda)}\right|<\dfrac{a}{2}$

for $z\in UC(\lambda,\alpha,\frac{1}{n})$. Then from Lemma 2.9, we obtain

$\displaystyle\
\mathcal{H}^{1}|_{\Gamma}\left(\mathcal{N}(h)\setminus\cup_{n=1}^{\infty}A_{n}\right)=0.$
(2.6)

It is clear that there exists $B_{n}\subset A_{n}$ with
$\mathcal{H}^{1}(B_{n})=0$ such that for each $\lambda_{0}\in A_{n}\setminus
B_{n}$, the following two properties hold.
(1) $\lambda_{0}$ is a Lebesgue point of $\chi_{A_{n}}$ and
$\frac{h(\lambda)}{L(\lambda)},$ where $\chi_{B}$ is the characteristic
function of the set $B,$ that is,

$\ \lim_{\delta\rightarrow
0}\dfrac{\mathcal{H}^{1}|_{\Gamma}(A_{n}^{c}\cap\mathbb{D}(\lambda_{0},\delta))}{\delta}=0$

and

$\ \lim_{\delta\rightarrow
0}\dfrac{\mathcal{H}^{1}|_{\Gamma}\left(\left\\{\lambda:\left|\frac{h(\lambda)}{L(\lambda)}-\frac{h(\lambda_{0})}{L(\lambda_{0})}\right|\geq\frac{a}{2}\right\\}\cap\mathbb{D}(\lambda_{0},\delta)\right)}{\delta}=0;$

(2) $\lambda_{0}$ satisfies (d), that is,

$\ \lim_{\delta\rightarrow
0}\dfrac{\mathcal{H}^{1}|_{\Gamma}\left(\\{\lambda:|\mathcal{C}(\nu)(\lambda)-\mathcal{C}(\nu)(\lambda_{0})|\geq\frac{a}{4}\\}\cap\mathbb{D}(\lambda_{0},\delta)\right)}{\delta}=0.$

Therefore,

$\ \lim_{\delta\rightarrow
0}\dfrac{\mathcal{H}^{1}|_{\Gamma}\left((A_{n}^{c}\cup\\{\lambda:|v^{+}(\nu,\Gamma)(\lambda)-v^{+}(\nu,\Gamma)(\lambda_{0})|\geq\frac{a}{2}\\})\cap\mathbb{D}(\lambda_{0},\delta)\right)}{\delta}=0.$

For $\epsilon>0$, there exists $\delta_{0}>0$ ($\delta_{0}<\frac{1}{2n}$) and
an open set $O_{\delta}\subset\mathbb{D}(\lambda_{0},\delta)$ for
$\delta<\delta_{0}$ such that

$\
\left(A_{n}^{c}\cup\left\\{\lambda:|v^{+}(\nu,\Gamma)(\lambda)-v^{+}(\nu,\Gamma)(\lambda_{0})|\geq\frac{a}{2}\right\\}\right)\cap\mathbb{D}(\lambda_{0},\delta)\subset
O_{\delta}$

and $\mathcal{H}^{1}|_{\Gamma}(O_{\delta})<\epsilon\delta$. Let
$E=\Gamma\cap(\overline{\mathbb{D}(\lambda_{0},\delta)}\setminus O_{\delta})$.
Denote

$\ B_{\delta}=\bigcup_{\lambda\in
E}\overline{UC\left(\lambda,\alpha,\frac{1}{n}\right)}.$

It is easy to check that $B_{\delta}$ is closed. By construction, we see that

$\ |\mathcal{C}(\nu)(z)-v^{+}(\nu,\Gamma)(\lambda_{0})|<a$

for $z\in B_{\delta}\setminus\Gamma$. Write

$\ U_{\Gamma}\cap\mathbb{D}(\lambda_{0},\delta)\setminus
B_{\delta}=\cup_{k=1}^{\infty}G_{k}$

where $G_{k}$ is a connected component. It is clear that
$diam(G_{k})\approx\mathcal{H}^{1}|_{\Gamma}(\overline{G_{k}}\cap\Gamma)$. Let
$I_{k}$ be the interior of $\overline{G_{k}}\cap\Gamma$ on $\Gamma$. Then
$\mathcal{H}^{1}|_{\Gamma}(\overline{G_{k}}\cap\Gamma)=\mathcal{H}^{1}|_{\Gamma}(I_{k})$
and $\cup_{k}I_{k}\subset O_{\delta}\cap\Gamma$. Hence, from Theorem 2.1 (2),
we get

$\ \gamma(U_{\Gamma}\cap\mathbb{D}(\lambda_{0},\delta)\setminus
B_{\delta})\leq A_{T}\sum_{k=1}^{\infty}\gamma(G_{k})\leq
A_{T}\sum_{k=1}^{\infty}diam(G_{k})\leq A_{T}C_{9}\epsilon\delta.$

This implies

$\displaystyle\ \lim_{\delta\rightarrow
0}\dfrac{\gamma\left(\\{\lambda:|\mathcal{C}(\nu)(\lambda)-v^{+}(\nu,\Gamma)(\lambda_{0})|>a\\}\cap\mathbb{D}(\lambda_{0},\delta)\cap
U_{\Gamma}\right)}{\delta}=0.$ (2.7)

So from (2.6), we have proved that there exists $\mathcal{Q}_{a}$ with
$\mathcal{H}^{1}(\mathcal{Q}_{a})=0$ such that for each
$\lambda_{0}\in\mathcal{N}(h)\setminus\mathcal{Q}_{a}$, (2.7) holds. Set
$\mathcal{Q}_{0}=\cup_{m=1}^{\infty}\mathcal{Q}_{\frac{1}{m}}$. Then (2.7)
holds for $\lambda_{0}\in\mathcal{N}(h)\setminus\mathcal{Q}_{0}$ and all
$a>0$. This proves (b). ∎

The proof of (c) is the same as (b). We point out if $\Gamma$ is a rotation of
a Lipschitz graph (with rotation angle $\beta$ and Lipschitz function $A$) at
$\lambda_{0}$, then

$\displaystyle\
v^{+}(\nu,\Gamma,\beta)(\lambda):=\mathcal{C}(\nu)(\lambda)+\frac{e^{-i\beta}h(\lambda)}{2L((\lambda-\lambda_{0})e^{-i\beta}+\lambda_{0})}.$
(2.8)

Similarly,

$\displaystyle\
v^{-}(\nu,\Gamma,\beta)(\lambda):=\mathcal{C}(\nu)(\lambda)-\frac{e^{-i\beta}h(\lambda)}{2L((\lambda-\lambda_{0})e^{-i\beta}+\lambda_{0})}.$
(2.9)

Hence, $v^{+}(\nu,\Gamma)(\lambda)=v^{+}(\nu,\Gamma,0)(\lambda)$ and
$v^{-}(\nu,\Gamma)(\lambda)=v^{-}(\nu,\Gamma,0)(\lambda).$

##  3\. Definition of the non-removable boundary and the removable set

From now on, we fix a compact subset $K\subset\mathbb{C},$ $\mu\in
M_{0}^{+}(K),$ $1\leq t<\infty,$ $\frac{1}{t}+\frac{1}{s}=1,$ and

$\ \Lambda=\\{g_{j}\\}\subset R^{t}(K,\mu)^{\perp}:=\left\\{g\in
L^{s}(\mu):~{}\int fgd\mu=0\text{ for }f\in\text{Rat}(K)\right\\}$

is a $L^{s}(\mu)$ norm dense subset.

The following definition generalizes [8, Definition 4.3].

######  Definition 3.1.

The non-removable boundary of zero density for $\Lambda$ is defined by

$\
\mathcal{F}_{0}(\Lambda)=\bigcap_{j=1}^{\infty}\\{z\in\mathcal{ZD}(\mu):~{}\mathcal{C}(g_{j}\mu)(z)=0\\}$

and the removable set of zero density for $\Lambda$ is defined by

$\
\mathcal{R}_{0}(\Lambda)=\bigcup_{j=1}^{\infty}\\{z\in\mathcal{ZD}(\mu):~{}\mathcal{C}(g_{j}\mu)(z)\neq
0\\}.$

For the non-zero density part of $\mu,$ we need to introduce further notation.
Let $\Gamma_{0}$ be a Lipschitz graph with rotation angle $\beta_{0},$ define

$\
\mathcal{Z}_{+}(\Lambda,\Gamma_{0})=\bigcap_{j=1}^{\infty}\\{z\in\Gamma_{0}:~{}v^{+}(g_{j}\mu,\Gamma_{0},\beta_{0})(z)=0\\},$
$\
\mathcal{Z}_{-}(\Lambda,\Gamma_{0})=\bigcap_{j=1}^{\infty}\\{z\in\Gamma_{0}:~{}v^{-}(g_{j}\mu,\Gamma_{0},\beta_{0})(z)=0\\},$
$\
\mathcal{N}_{+}(\Lambda,\Gamma_{0})=\bigcup_{j=1}^{\infty}\\{z\in\Gamma_{0}:~{}v^{+}(g_{j}\mu,\Gamma_{0},\beta_{0})(z)\neq
0\\},$

and

$\
\mathcal{N}_{-}(\Lambda,\Gamma_{0})=\bigcup_{j=1}^{\infty}\\{z\in\Gamma_{0}:~{}v^{-}(g_{j}\mu,\Gamma_{0},\beta_{0})\neq
0\\}.$

The following lemma, which describes the decomposition of $\mu\in
M_{0}^{+}(K),$ is important for our definitions.

######  Lemma 3.2.

Let $K$ be a compact subset and $\mu\in M_{0}^{+}(K).$ Then there is a
sequence of Lipschitz functions $A_{n}:\mathbb{R}\rightarrow\mathbb{R}$ and
such that their graphs $\Gamma_{n}$ with rotation angles $\beta_{n}$ satisfy
the following properties:

(1) Let $\Gamma=\cup_{n}\Gamma_{n}$. Then
$\mu=h\mathcal{H}^{1}|_{\Gamma}+\mu_{s}$ is the Radon-Nikodym decomposition
with respect to $\mathcal{H}^{1}|_{\Gamma}$, where $h\in
L^{1}(\mathcal{H}^{1}|_{\Gamma})$ and $\mu_{s}\perp\mathcal{H}^{1}|_{\Gamma};$

(2) $\mathcal{ND}(\mu)\approx\mathcal{N}(h),~{}\gamma-a.a.;$

(3) $\Theta_{\mu_{s}}(z)=0,~{}\gamma-a.a..$

###### Proof.

From Lemma 2.6 (b), we have

$\ \mathcal{ND}(\mu)=\bigcup_{n}\mathcal{ND}(\mu,n)\cup\mathcal{Q}_{\mu}.$

Thus, we find $h$ such that
$\mu|_{\mathcal{ND}(\mu,n)}=h\mathcal{H}^{1}|_{\mathcal{ND}(\mu,n)}$ and
$\mathcal{H}^{1}(\mathcal{ND}(\mu,n))<\infty.$

From [16, Theorem 1.26], $\mathcal{ND}(\mu,n)=E_{r}\cup E_{u},$ where $E_{r}$
is rectifiable and $E_{u}$ is purely unrectifiable. From David’s Theorem (see
[9] or [16, Theorem 7.2]), we see that $\gamma(E_{u})=0.$ Applying [16,
Proposition 4.13], we find a sequence of (rotated) Lipschitz graphs
$\\{\Gamma_{nm}\\}$ and $E_{1}$ with $\mathcal{H}^{1}(E_{1})=0$ such that
$E_{r}\subset E_{1}\cup\cup_{m=1}^{\infty}\Gamma_{nm}.$ Let
$\mathcal{Q}=E_{u}\cup E_{1}.$ Clearly, $\gamma(\mathcal{Q})=0.$ Hence, there
exists a sequence of $\\{\Gamma_{n}\\}$ such that (1), (2), and (3) hold. ∎

We are now ready to define the non-removable boundary and removable set for
non-zero density portion of $\mu,$ which extends [8, Definition 4.6].

######  Definition 3.3.

Let $\Gamma_{n}$ and $h$ be as in Lemma 3.2. Define the upper non-removable
boundary for the non-zero density portion of $\mu$ as

$\
\mathcal{F}_{+}(\Lambda):=\bigcup_{n=1}^{\infty}\mathcal{Z}_{+}(\Lambda,\Gamma_{n})\cap\mathcal{ND}(\mu),$

the lower non-removable boundary for the non-zero density portion of $\mu$ as

$\
\mathcal{F}_{-}(\Lambda):=\bigcup_{n=1}^{\infty}\mathcal{Z}_{-}(\Lambda,\Gamma_{n})\cap\mathcal{ND}(\mu),$

and the removable set for the non-zero density portion of $\mu$ as

$\
\mathcal{R}_{1}(\Lambda):=\bigcup_{n=1}^{\infty}\left(\mathcal{N}_{+}(\Lambda,\Gamma_{n})\cap\mathcal{N}_{-}(\Lambda,\Gamma_{n})\right)\cap\mathcal{ND}(\mu),$

Finally, we have the following definition.

######  Definition 3.4.

The non-removable boundary for $\Lambda$ is defined as

$\
\mathcal{F}(\Lambda)=\mathcal{F}_{0}(\Lambda)\cup\mathcal{F}_{+}(\Lambda)\cup\mathcal{F}_{-}(\Lambda)$

and the removable set for $\Lambda$ is defined as

$\
\mathcal{R}(\Lambda):=\mathcal{R}_{0}(\Lambda)\cup\mathcal{R}_{1}(\Lambda).$

In the remaining section, we discuss a characterization of
$\mathcal{F}(\Lambda)$ and $\mathcal{R}(\Lambda)$ which implies Definition 3.4
does not depend on the choices of $\\{\Gamma_{n}\\}$, up to a set of zero
analytic capacity. First we need the following lemma that will also be used
later.

######  Lemma 3.5.

Suppose that $\\{g_{n}\\}\subset L^{1}(\mu)$ and $E$ is a compact subset with
$\gamma(E)>0$. Then there exists $\eta\in M_{0}^{+}(E)$ satisfying:

(1) $\eta$ is of $1$-linear growth,
$\|\mathcal{C}_{\epsilon}(\eta)\|_{L^{\infty}(\mathbb{C})}\leq 1$ for all
$\epsilon>0,$ and $\gamma(E)\leq C_{10}\|\eta\|$;

(2) $\mathcal{C}_{*}(g_{n}\mu)\in L^{\infty}(\eta)$;

(3) there exists a subsequence $f_{k}(z)=\mathcal{C}_{\epsilon_{k}}(\eta)(z)$
such that $f_{k}$ converges to $f\in L^{\infty}(\mu)$ in weak-star topology,
and $f_{k}(\lambda)$ converges to $f(\lambda)=\mathcal{C}(\eta)(\lambda)$
uniformly on any compact subset of $E^{c}$ as $\epsilon_{k}\rightarrow 0$.
Moreover for $n\geq 1,$

$\displaystyle\ \int
f(z)g_{n}(z)d\mu(z)=-\int\mathcal{C}(g_{n}\mu)(z)d\eta(z),$ (3.1)

and for $\lambda\in E^{c}$,

$\displaystyle\
\int\dfrac{f(z)-f(\lambda)}{z-\lambda}g_{n}(z)d\mu(z)=-\int\mathcal{C}(g_{n}\mu)(z)\dfrac{d\eta(z)}{z-\lambda}.$
(3.2)

###### Proof.

From Lemma 2.5, we find $E_{1}\subset E$ such that $\gamma(E\setminus
E_{1})<\frac{\gamma(E)}{2A_{T}}$ and $\mathcal{C}_{*}(g_{n}\mu)(z)\leq
M_{n}<\infty$ for $z\in E_{1}$. Using Theorem 2.1 (2), we get
$\gamma(E_{1})\geq\frac{1}{A_{T}}\gamma(E)-\gamma(E\setminus
E_{1})\geq\frac{1}{2A_{T}}\gamma(E).$ Using Theorem 2.1 (1) and Proposition
2.2 (1), there exists $\eta\in M_{0}^{+}(E_{1})$ satisfying (1). So (2) holds.
Moreover, using Proposition 2.2 again, we may assume $\eta=w\eta_{1},$ where
$\eta_{1}$ is of $1$-linear growth, $0\leq w\leq 1,$ and $N_{2}(\eta_{1})\leq
1.$ Hence, from Theorem 2.1 (1) and Proposition 2.2 (3), we conclude a
$\gamma$ zero set is also a zero set of $\eta.$ Clearly,

$\displaystyle\
\int\mathcal{C}_{\epsilon}(\eta)(z)g_{n}d\mu=-\int\mathcal{C}_{\epsilon}(g_{n}\mu)(z)d\eta$
(3.3)

for $n\geq 1$. We can choose a sequence
$f_{k}(\lambda)=\mathcal{C}_{\epsilon_{k}}(w\eta)(\lambda)$ that converges to
$f$ in $L^{\infty}(\mu)$ weak-star topology and $f_{k}(\lambda)$ uniformly
tends to $f(\lambda)$ on any compact subset of $E^{c}$. On the other hand,
$|\mathcal{C}_{\epsilon_{k}}(g_{n}\mu)(z)|\leq M_{n},~{}\eta|_{E}-a.a.$ and by
Corollary 2.3,
$\lim_{k\rightarrow\infty}\mathcal{C}_{\epsilon_{k}}(g_{n}\mu)(z)=\mathcal{C}(g_{n}\mu)(z),~{}\eta-a.a.$.
We apply the Lebesgue dominated convergence theorem to the right hand side of
(3.3) and get (3.1) for $n\geq 1$. For (3.2), let $\lambda\notin E$ and
$d=\text{dist}(\lambda,E)$, for $z\in\mathbb{D}(\lambda,\frac{d}{2})$ and
$\epsilon<\frac{d}{2}$, we have

$\
\left|\dfrac{\mathcal{C}_{\epsilon}(\eta)(z)-f(\lambda)}{z-\lambda}\right|\leq\left|\mathcal{C}_{\epsilon}\left(\dfrac{\eta(s)}{s-\lambda}\right)(z)\right|\leq\dfrac{2}{d^{2}}\|\eta\|.$

For $z\notin\mathbb{D}(\lambda,\frac{d}{2})$ and $\epsilon<\frac{d}{2}$,

$\
\left|\dfrac{\mathcal{C}_{\epsilon}(\eta)(z)-f(\lambda)}{z-\lambda}\right|\leq\dfrac{4}{d}.$

Thus, we can replace the above proof for the measure
$\frac{\eta(s)}{s-\lambda}$. In fact, we can choose a subsequence
$\\{\mathcal{C}_{\epsilon_{k_{j}}}(\eta)\\}$ such that
$e_{k_{j}}(z)=\frac{\mathcal{C}_{\epsilon_{k_{j}}}(\eta)(z)-f(\lambda)}{z-\lambda}$
converges to $e(z)$ in weak-star topology. Clearly,
$(z-\lambda)e_{k_{j}}(z)+f(\lambda)=\mathcal{C}_{\epsilon_{k_{j}}}(\eta)(z)$
converges to $(z-\lambda)e(z)+f(\lambda)=f(z)$ in weak-star topology. On the
other hand, 3.3 becomes

$\displaystyle\
\int\mathcal{C}_{\epsilon_{k_{j}}}(\dfrac{\eta(s)}{s-\lambda})(z)g_{n}d\mu=-\int\mathcal{C}_{\epsilon_{k_{j}}}(g_{n}\mu)(z)\dfrac{d\eta(z)}{z-\lambda}$
(3.4)

and for $\epsilon_{k_{j}}<\frac{d}{2}$, we have

$\displaystyle\left|\mathcal{C}_{\epsilon_{k_{j}}}(\dfrac{\eta(s)}{s-\lambda})(z)-e_{k_{j}}(z)\right|\
\leq\begin{cases}0,&z\in\mathbb{D}(\lambda,\frac{d}{2}),\\\
\dfrac{2}{d^{2}}\eta(\mathbb{D}(z,\epsilon_{k_{j}})),&z\notin\mathbb{D}(\lambda,\frac{d}{2}),\end{cases}$
(3.5)

which goes to zero as $\epsilon_{k_{j}}\rightarrow 0$. Combining (3.4), (3.5),
and Lebesgue dominated convergence theorem, we prove the equation (3.2). (3)
is proved. ∎

For a Lipschitz graph $\Gamma_{0}$ with rotation angle $\beta_{0},$ define

$\
\mathcal{N}_{0}(\Lambda,\Gamma_{0})=\bigcup_{j=1}^{\infty}\\{z\in\Gamma_{0}:~{}{\mathcal{C}}(g_{j}\mu)(z)\neq
0\\}.$

######  Lemma 3.6.

Let $\Gamma_{n}$ and $h$ be as in Lemma 3.2. Then, for $n\geq 1,$

$\
\mathcal{N}_{+}(\Lambda,\Gamma_{n})\cup\mathcal{N}_{-}(\Lambda,\Gamma_{n})\subset\mathcal{N}_{0}(\Lambda,\Gamma_{n}),~{}\mathcal{H}^{1}|_{\Gamma_{n}}-a.a..$

###### Proof.

Without loss of generality, we assume $n=1$ and $\beta_{1}=0.$ Suppose there
exists a compact subset
$E\subset\mathcal{N}_{+}(\Lambda,\Gamma_{1})\cup\mathcal{N}_{-}(\Lambda,\Gamma_{1})\setminus\mathcal{N}_{0}(\Lambda,\Gamma_{1})$
such that $\mathcal{H}^{1}(E)>0.$ Then
${\mathcal{C}}(g_{j}\mu)(z)=0,~{}\mathcal{H}^{1}|_{E}-a.a.$ and

$\displaystyle\
v^{+}(g_{j}\mu,\Gamma_{1})(z)=\dfrac{(g_{j}h)(z)}{2L(z)},~{}v^{-}(g_{j}\mu,\Gamma_{1})(z)=-\dfrac{(g_{j}h)(z)}{2L(z)},~{}\mathcal{H}^{1}|_{E}-a.a.$
(3.6)

for all $j\geq 1.$ Let $\eta$ and $f$ be from Lemma 3.5 for $\Lambda$ and $E$
as $\gamma(E)>0$ by (2.3). We may assume $\eta=w\mathcal{H}^{1}|_{E},$ where
$0\leq w(z)\leq 1$ on $E,$ since $\eta$ is of $1$-linear growth. From Lemma
3.5 (3), we see $f\in R^{t,\infty}(K,\mu)$ as $\Lambda$ is dense in
$R^{t}(K,\mu)^{\perp}$ and

$\displaystyle\
{\mathcal{C}}\eta(\lambda){\mathcal{C}}(g_{j}\mu)(\lambda)={\mathcal{C}}(fg_{j}\mu)(\lambda),~{}\gamma|_{E^{c}}-a.a.,~{}\text{
for }j\geq 1.$ (3.7)

Let $\\{r_{n}\\}\subset\text{Rat}(K)$ such that
$\|r_{n}-f\|_{L^{t}(\mu)}\rightarrow 0.$ Clearly,

$\
\mathcal{C}(r_{n}g_{j}\mu)(z)=r_{n}(z)\mathcal{C}(g_{j}\mu)(z)=0,~{}\eta-a.a.$

since a $\gamma|_{E}$ zero set is also a zero set of $\eta$ by (2.3). By
Theorem 2.1 (3), we get

$\displaystyle\ \eta\\{|\mathcal{C}(fg_{j}\mu)|>\epsilon\\}=$
$\displaystyle\eta\\{|\mathcal{C}((r_{n}-f)g_{j}\mu)|>\epsilon\\}$
$\displaystyle\ \leq$
$\displaystyle\eta\\{\mathcal{C}_{*}((r_{n}-f)g_{j}\mu)>\epsilon\\}$
$\displaystyle\ \leq$
$\displaystyle\dfrac{C_{11}\|r_{n}-f)\|\|g_{j}\|}{\epsilon}\rightarrow 0\text{
as }n\rightarrow 0.$

Therefore, $\mathcal{C}(fg_{j}\mu)(z)=0,~{}\eta-a.a.$ and we have

$\displaystyle\
v^{+}(fg_{j}\mu,\Gamma_{1})(z)=\dfrac{(fg_{j}h)(z)}{2L(z)},~{}v^{-}(fg_{j}\mu,\Gamma_{1})(z)=-\dfrac{(fg_{j}h)(z)}{2L(z)},~{}\mathcal{\eta}-a.a.$
(3.8)

for $j\geq 1.$ Apply Theorem 2.11 and Lemma 2.9 to
${\mathcal{C}}\eta(\lambda)$ for (3.7), we get

$\displaystyle\
v^{+}(w\mathcal{H}^{1},\Gamma_{1})(z)v^{+}(g_{j}\mu,\Gamma_{1})(z)=$
$\displaystyle v^{+}(fg_{j}\mu,\Gamma_{1})(z),~{}\eta-a.a.,$ $\displaystyle\
v^{-}(w\mathcal{H}^{1},\Gamma_{1})(z)v^{-}(g_{j}\mu,\Gamma_{1})(z)=$
$\displaystyle v^{-}(fg_{j}\mu,\Gamma_{1})(z),~{}\eta-a.a.$

Combining with (3.6) and (3.8), we have

$\ g_{j}(z)h(z)=0,~{}\eta-a.a.\text{ for }j\geq 1,$

which implies

$\
v^{+}(g_{j}\mu,\Gamma_{1})(z)=v^{-}(g_{j}\mu,\Gamma_{1})(z)=0,~{}\eta-a.a.\text{
for }j\geq 1.$

This is a contradiction. The lemma is proved. ∎

Define

$\ \mathcal{E}_{N}(\Lambda)=\left\\{\lambda:~{}\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(g_{j}\mu)(\lambda)\text{ exists, }\max_{1\leq j\leq
N}|\mathcal{C}(g_{j}\mu)(\lambda)|\leq\frac{1}{N}\right\\}.$

######  Theorem 3.7.

There is a subset $\mathcal{Q}\subset\mathbb{C}$ with $\gamma(\mathcal{Q})=0$
such that if $\lambda\in\mathbb{C}\setminus\mathcal{Q}$, then
$\lambda\in\mathcal{F}(\Lambda)$ if and only if

$\displaystyle\ \underset{\delta\rightarrow
0}{\overline{\lim}}\dfrac{\gamma(\mathbb{D}(\lambda,\delta)\cap\mathcal{E}_{N}(\Lambda))}{\delta}>0$
(3.9)

for all $N\geq 1$. Consequently, $\mathcal{F}(\Lambda)$ and
$\mathcal{R}(\Lambda)$ do not depend on the choices of $\\{\Gamma_{n}\\}$ (as
in Lemma 3.2), up to a set of zero analytic capacity.

###### Proof.

We first prove that there exists $\mathcal{Q}_{1}$ with
$\gamma(\mathcal{Q}_{1})=0$ such that if
$\lambda_{0}\in\mathcal{ZD}(\mu)\setminus\mathcal{Q}_{1}$, then
$\lambda_{0}\in\mathcal{F}(\Lambda)$ if and only if $\lambda_{0}$ satisfies
(3.9).

From Definition 3.1, we get
$\mathcal{ZD}(\mu)\approx\mathcal{R}_{0}(\Lambda)\cup\mathcal{F}_{0}(\Lambda),~{}\gamma-a.a.$.
There exists $\mathcal{Q}_{1}$ with $\gamma(\mathcal{Q}_{1})=0$ such that for
$\lambda_{0}\in\mathcal{ZD}(\mu)\setminus\mathcal{Q}_{1}$,
$\Theta_{g_{j}\mu}(\lambda_{0})=0$,
$\mathcal{C}(g_{j}\mu)(\lambda_{0})=\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(g_{j}\mu)(\lambda_{0})$ exists, and
$\mathcal{C}(g_{j}\mu)(z)$ is $\gamma$-continuous at $\lambda_{0}$ (Lemma 2.8)
for all $j\geq 1$.

If $\lambda_{0}\in\mathcal{R}_{0}(\Lambda)$, then there exists $j_{0}$ such
that $\mathcal{C}(g_{j_{0}}\mu)(\lambda_{0})\neq 0$. Let
$\epsilon_{0}=\frac{1}{2}|\mathcal{C}(g_{j_{0}}\mu)(\lambda_{0})|$, we obtain
that, for $N>N_{0}:=\max(j_{0},\frac{1}{\epsilon_{0}}+1)$,

$\
\mathcal{E}_{N}(\Lambda)\subset\\{|\mathcal{C}(g_{j_{0}}\mu)(z)-\mathcal{C}(g_{j_{0}}\mu)(\lambda_{0})|>\epsilon_{0}\\}.$

Therefore, by Lemma 2.8, for $N>N_{0}$,

$\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap\mathcal{E}_{N}(\Lambda))}{\delta}=0.$

Thus, $\lambda_{0}$ does not satisfy (3.9).

Now for $\lambda_{0}\in\mathcal{F}_{0}(\Lambda)$,
$\mathcal{C}(g_{j}\mu)(\lambda_{0})=0$ for all $j\geq 1$. Using Lemma 2.8 and
Theorem 2.1 (2), we get

$\ \begin{aligned} \ &\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\setminus\mathcal{E}_{N}(\Lambda))}{\delta}\\\
\ \leq&A_{T}\sum_{j=1}^{N}\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap\\{|\mathcal{C}(g_{j}\mu)(z)-\mathcal{C}(g_{j}\mu)(\lambda_{0})|\geq\frac{1}{N}\\})}{\delta}\\\
\ =&0.\ \end{aligned}$

Hence, $\lambda_{0}$ satisfies (3.9).

We now prove that there exists $\mathcal{Q}_{2}$ with
$\gamma(\mathcal{Q}_{2})=0$ such that if
$\lambda_{0}\in\mathcal{ND}(\mu)\setminus\mathcal{Q}_{2}$, then
$\lambda_{0}\in\mathcal{F}(\Lambda)$ if and only if $\lambda_{0}$ satisfies
(3.9).

From Definition 3.3, we get
$\mathcal{ND}(\mu)\approx\mathcal{R}_{1}(\Lambda)\cup(\mathcal{F}_{+}(\Lambda)\cup\mathcal{F}_{-}(\Lambda)),~{}\gamma-a.a.$.
There exists $\mathcal{Q}_{2}$ with $\gamma(\mathcal{Q}_{2})=0$ such that for
$\lambda_{0}\in\mathcal{ND}(\mu)\cap\Gamma_{n}\setminus\mathcal{Q}_{2}$,
$v^{0}(g_{j}\mu,\Gamma_{n})(\lambda_{0})=\mathcal{C}(g_{j}\mu)(\lambda_{0})=\lim_{\epsilon\rightarrow
0}\mathcal{C}_{\epsilon}(g_{j}\mu)(\lambda_{0})$,
$v^{+}(g_{j}\mu,\Gamma_{n},\beta_{n})(\lambda_{0})$, and
$v^{-}(g_{j}\mu,\Gamma_{n},\beta_{n})(\lambda_{0})$ exist for all $j,n\geq 1$
and Theorem 2.11 (b), (c), and (d) hold. Fix $n=1$ and without loss of
generality, we assume $\beta_{1}=0.$

If $\lambda_{0}\in\mathcal{R}_{1}(\Lambda)$, then there exist integers
$j_{0}$, $j_{1}$, and $j_{2}$ such that
$v^{0}(g_{j_{0}}\mu,\Gamma_{1})(\lambda_{0})\neq 0$ by Lemma 3.6 (in fact, we
can let $j_{0}=j_{1}$ or $j_{0}=j_{2}$),
$v^{+}(g_{j_{1}},\Gamma_{1})(\lambda_{0})\neq 0$, and
$v^{-}(g_{j_{2}}\mu,\Gamma_{1})(\lambda_{0})\neq 0$. Set

$\
\epsilon_{0}=\dfrac{1}{2}\min(|v^{0}(g_{j_{0}}\mu,\Gamma_{1})(\lambda_{0})|,|v^{+}(g_{j_{1}},\Gamma_{1})(\lambda_{0})|,|v^{-}(g_{j_{2}}\mu,\Gamma_{1})(\lambda_{0})|),$

then for $N>N_{0}:=\max(j_{0},j_{1},j_{2},\frac{1}{\epsilon_{0}}+1)$,

$\ \Gamma_{1}\cap\mathcal{E}_{N}(\Lambda)\subset
D:=\Gamma_{1}\cap\\{|\mathcal{C}(g_{j_{0}}\mu)(z)-v^{0}(g_{j_{0}}\mu,\Gamma_{1})(\lambda_{0})|\geq\epsilon_{0}\\},$
$\ U_{\Gamma_{1}}\cap\mathcal{E}_{N}(\Lambda)\subset
E:=U_{\Gamma_{1}}\cap\\{|\mathcal{C}(g_{j_{1}}\mu)(z)-v^{+}(g_{j_{1}}\mu,\Gamma_{1})(\lambda_{0})|\geq\epsilon_{0}\\},$

and

$\ L_{\Gamma_{1}}\cap\mathcal{E}_{N}(\Lambda)\subset
F:=L_{\Gamma_{1}}\cap\\{|\mathcal{C}(g_{j_{2}}\mu)(z)-v^{-}(g_{j_{2}}\mu,\Gamma_{1})(\lambda_{0})|\geq\epsilon_{0}\\}.$

Therefore, using Theorem 2.1 (2) and Theorem 2.11, we get for $N>N_{0}$,

$\displaystyle\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap\mathcal{E}_{N}(\Lambda))}{\delta}$
$\displaystyle\ \leq$ $\displaystyle A_{T}\left(\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap
D)}{\delta}+\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap
E)}{\delta}+\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap F)}{\delta}\right)$
$\displaystyle\ =$ $\displaystyle 0.$

Hence, $\lambda_{0}$ does not satisfy (3.9).

For
$\lambda_{0}\in(\mathcal{F}_{+}(\Lambda)\cup\mathcal{F}_{-}(\Lambda))\cap\Gamma_{1}$,
we may assume that
$\lambda_{0}\in\mathcal{Z}_{+}(\Lambda,\Gamma_{1})\subset\Gamma_{1}$. Using
Theorem 2.1 (2) and Theorem 2.11, we get
($v^{+}(g_{j}\mu,\Gamma_{1})(\lambda_{0})=0$)

$\ \begin{aligned} \ &\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap
U_{\Gamma_{1}}\setminus\mathcal{E}_{N}(\Lambda))}{\delta}\\\ \
\leq&A_{T}\sum_{j=1}^{N}\lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap
U_{\Gamma_{1}}\cap\\{|\mathcal{C}(g_{j}\mu)(z)-v^{+}(g_{j}\mu,\Gamma_{1})(\lambda_{0})|\geq\frac{1}{N}\\})}{\delta}\\\
\ \ =&0.\ \end{aligned}$

This implies

$\ \underset{\delta\rightarrow
0}{\overline{\lim}}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap\mathcal{E}_{N}(\Lambda)}{\delta}\geq\underset{\delta\rightarrow
0}{\overline{\lim}}\dfrac{\gamma(\mathbb{D}(\lambda_{0},\delta)\cap
U_{\Gamma_{1}}\cap\mathcal{E}_{N}(\Lambda))}{\delta}>0.$

Hence, $\lambda_{0}$ satisfies (3.9).

Finally, (3.9) does not depend on choices of $\\{\Gamma_{n}\\}$, therefore,
$\mathcal{F}(\Lambda)$ and $\mathcal{R}(\Lambda)$ are independent of choices
of $\\{\Gamma_{n}\\}$ up to a set of zero analytic capacity. ∎

##  4\. Building block functions and uniqueness of $\mathcal{F}(\Lambda)$ and
$\mathcal{R}(\Lambda)$

The following is our main lemma, which extends [8, Lemma 5.1] to $\Lambda.$

######  Lemma 4.1.

Let $E_{1}\subset\mathcal{F}(\Lambda)$ be a compact subset with
$\gamma(E_{1})>0$. Then there exists $f\in R^{t,\infty}(K,\mu),$ $\eta\in
M_{0}^{+}(E_{1}),$ and $\|\mathcal{C}_{\epsilon}(\eta)\|\leq C_{12}$ such that
$f(z)=\mathcal{C}(\eta)(z)$ for $z\in\mathbb{C}_{\infty}\setminus E_{1}$,

$\ \|f\|_{L^{\infty}(\mu)}\leq
C_{12},~{}f(\infty)=0,~{}f^{\prime}(\infty)=-\gamma(E_{1}),$

and

$\displaystyle\
\mathcal{C}(\eta)(z)\mathcal{C}(g_{j}\mu)(z)=\mathcal{C}(fg_{j}\mu)(z),~{}\gamma|_{E_{1}^{c}}-a.a.\text{
for }j\geq 1.$ (4.1)

From the semi-additivity of $\gamma$ (see Theorem 2.1 (2)), we have

$\ \gamma(E_{1})\leq
A_{T}(\gamma(E_{1}\cap\mathcal{F}_{0}(\Lambda))+\gamma(E_{1}\cap\mathcal{F}_{+}(\Lambda))+\gamma(E_{1}\cap\mathcal{F}_{-}(\Lambda))).$

Hence,

$\
\max(\gamma(E_{1}\cap\mathcal{F}_{0}(\Lambda)),~{}\gamma(E_{1}\cap\mathcal{F}_{+}(\Lambda)),~{}\gamma(E_{1}\cap\mathcal{F}_{-}(\Lambda)))\geq\dfrac{1}{3A_{T}}\gamma(E_{1}).$

Therefore, we shall prove Lemma 4.1 assuming
$E_{1}\subset\mathcal{F}_{0}(\Lambda),$
$E_{1}\subset\mathcal{F}_{+}(\Lambda),$ or
$E_{1}\subset\mathcal{F}_{-}(\Lambda).$ In the following proofs, we fix the
measure $\eta=\eta_{1}$ and the function $f=f_{1}$ from Lemma 3.5 for
$\\{g_{n}\\}=\Lambda$ and $E=E_{1}$.

###### Proof.

(Lemma 4.1 assuming $E_{1}\subset\mathcal{F}_{0}(\Lambda)$): From (3.1) and
(3.2), we see that $f_{1}\in R^{t,\infty}(K,\mu)$ and
$f_{1}(\lambda)\mathcal{C}(g_{j}\mu)(\lambda)=\mathcal{C}(f_{1}g_{j}\mu)(\lambda,~{}\gamma|_{E_{1}^{c}}-a.a.$
for $j\geq 1.$ Set

$\ f=\dfrac{f_{1}}{\|\eta_{1}\|}\gamma(E_{1})\text{ and
}\eta=\dfrac{\eta_{1}}{\|\eta_{1}\|}\gamma(E_{1}).$

Then $f$ and $\eta$ satisfy the properties of the lemma. ∎

###### Proof.

(Lemma 4.1 assuming $E_{1}\subset\mathcal{F}_{+}(\Lambda)$): Using Corollary
2.4, we assume that there exists a positive integer $n_{0}$ such that
$E_{1}\subset\bigcup_{n=1}^{n_{0}}\Gamma_{n}.$ Put
$E_{1}=\cup_{n=1}^{n_{0}}F_{n},$ where $F_{n}\subset\Gamma_{n}$ and $F_{n}\cap
F_{m}=\emptyset$ for $n\neq m.$ Since $\eta_{1}$ is of linear growth, we can
set $\eta_{1}=\sum_{n=1}^{n_{0}}w_{n}(z)\mathcal{H}^{1}|_{\Gamma_{n}},$ where
$w_{n}$ is supported on $F_{n}.$ The function

$\
f_{2}(z)=f_{1}(z)-\frac{1}{2}\sum_{n=1}^{n_{0}}e^{-i\beta_{n}}L((z-z_{n})e^{-i\beta_{n}}+z_{n})^{-1}w_{n}(z).$

is the non-tangential limit of $f_{1}$ from the bottom on $E_{1}$ (Lemma 2.9).
Thus, $f_{2},w_{n}\in L^{\infty}(\mu)$ and $\|f_{2}\|_{L^{\infty}(\mu)}\leq
C_{12}.$ From (3.1), we get

$\displaystyle\begin{aligned} \ \int
f_{2}g_{j}d\mu=&-\int\mathcal{C}(g_{j}\mu)d\eta_{1}-\frac{1}{2}\sum_{n=1}^{n_{0}}\int_{F_{n}}e^{-i\beta_{n}}L((z-z_{n})e^{-i\beta_{n}}+z_{n})^{-1}g_{j}hd\eta_{1}\\\
\
=&-\sum_{n=1}^{n_{0}}\int_{F_{n}}v^{+}(g_{j}\mu,\Gamma_{n},\beta_{n})d\eta_{1}.\end{aligned}$
(4.2)

Similarly, for $\lambda\in E_{1}^{c}$ and by (3.2), we have

$\displaystyle\
\int\dfrac{f_{2}(z)-f_{2}(\lambda)}{z-\lambda}g_{j}(z)d\mu(z)=-\sum_{n=1}^{n_{0}}\int_{F_{n}}v^{+}(g_{j}\mu,\Gamma_{n},\beta_{n})(z)\dfrac{d\eta_{1}(z)}{z-\lambda}.$
(4.3)

Since $v^{+}(g_{j}\mu,\Gamma_{n},\beta_{n})(z)=0,~{}z\in
F_{n},~{}\mathcal{H}^{1}|_{\Gamma_{n}}-a.a.,$ by (4.2), we get $f_{2}\in
R^{t,\infty}(K,\mu).$ Similarly, from (4.3), we see that
$\frac{f_{2}(z)-f_{2}(\lambda)}{z-\lambda}\in R^{t,\infty}(K,\mu)$ and
$f_{2}(\lambda)\mathcal{C}(g_{j}\mu)(\lambda)=\mathcal{C}((f_{2}g_{j}\mu)(\lambda)$
for $\lambda\in E_{1}^{c},~{}\gamma-a.a.$ Set

$\ f=\dfrac{f_{2}}{\|\eta_{1}\|}\gamma(E_{1})\text{ and
}\eta=\dfrac{\eta_{1}}{\|\eta_{1}\|}\gamma(E_{1}).$

Then $f$ and $\eta$ satisfy the properties of the lemma. ∎

The proof of Lemma 4.1 for $E_{1}\subset\mathcal{F}_{-}(\Lambda)$ is the same
as that for $E_{1}\subset\mathcal{F}_{+}(\Lambda)$ if we modify the definition
of $f_{2}$ by the following

$\
f_{2}(z)=f_{1}(z)+\frac{1}{2}\sum_{n=1}^{n_{0}}e^{-i\beta_{n}}L((z-z_{n})e^{-i\beta_{n}}+z_{n})^{-1}w_{n}(z).$

We now show the properties of $g_{j}\mu$ on $\mathcal{F}(\Lambda)$ are
preserved for all annihilating measures $g\mu$ for $g\in
R^{t}(K,\mu)^{\perp}.$

######  Theorem 4.2.

Let $\Gamma_{n}$, $\Gamma$, and $d\mu=hd\mathcal{H}^{1}|_{\Gamma}+d\mu_{s}$ be
as in Lemma 3.2. Then for $g\in R^{t}(K,\mu)^{\perp},$

$\displaystyle\
\mathcal{C}(g\mu)(\lambda)=0,~{}\gamma|_{\mathcal{F}_{0}(\Lambda)}-a.a.,$
(4.4) $\displaystyle\
v^{+}(g\mu,\Gamma_{n},\beta_{n})(\lambda)=0,~{}\gamma|_{\mathcal{F}_{+}(\Lambda)\cap\Gamma_{n}}-a.a.~{}\text{for}~{}n\geq
1,$ (4.5)

and

$\displaystyle\
v^{-}(g\mu,\Gamma_{n},\beta_{n})(\lambda)=0,~{}\gamma|_{\mathcal{F}_{-}(\Lambda)\cap\Gamma_{n}}-a.a.~{}\text{for}~{}n\geq
1.$ (4.6)

###### Proof.

Equation (4.4): Assume that there exists a compact subset
$E_{1}\subset\mathcal{F}_{0}(\Lambda)$ such that $\gamma(E_{1})>0$ and

$\displaystyle\ Re(\mathcal{C}(g\mu))>0,~{}\lambda\in E_{1}.$ (4.7)

Using Lemma 2.5, we assume that $\mathcal{C}_{*}(g\mu)(\lambda)<M_{1}$ for
$\lambda\in E_{1}$. Let $f\in R^{t,\infty}(K,\mu)$ and $\eta\in
M_{0}^{+}(E_{1})$ be as in Lemma 4.1. By (3.1), we get

$\ \int Re(\mathcal{C}(g\mu)(z))d\eta(z)=-Re\left(\int
f(z)g(z)d\mu(z)\right)=0,$

which implies $Re(\mathcal{C}(g\mu)(z))=0,~{}\eta-a.a..$ This contradicts to
(4.7).

Equation (4.5): Assume that there exists a compact subset
$E_{1}\subset\mathcal{F}_{+}(\Lambda)\cap\Gamma_{1}$ such that
$\gamma(E_{1})>0$ and

$\displaystyle\ Re(v^{+}(g\mu,\Gamma_{1},\beta_{1})(z))>0,~{}\lambda\in
E_{1}.$ (4.8)

Using Lemma 2.5, we assume that $\mathcal{C}_{*}(g\mu)(\lambda)<M_{1}$ for
$\lambda\in E_{1}$. Let $f\in R^{t,\infty}(K,\mu)$ and $\eta\in
M_{0}^{+}(E_{1})$ be as in Lemma 4.1. Similar to (4.2), we see that

$\ \int Re(v^{+}(g\mu,\Gamma_{1},\beta_{1})(z))d\eta(z)=-Re\left(\int
f(z)g(z)d\mu(z)\right)=0,$

which implies $Re(v^{+}(g\mu,\Gamma_{1},\beta_{1})(z))=0,~{}\eta-a.a..$ This
contradicts to (4.8).

The proof of the equation (4.6) is the same as above. ∎

As a simple application of Theorem 4.2 (4.4) and the fact that
${\mathfrak{m}}(\mathcal{F}_{+}(\Lambda)\cup\mathcal{F}_{-}(\Lambda))=0$, we
have the corollary below.

######  Corollary 4.3.

For $g\perp R^{t}(K,\mu)$,
$\mathcal{C}(g\mu)(z)=0,~{}{\mathfrak{m}}|_{\mathcal{F}(\Lambda)}-a.a.$.

The following corollary is a direct outcome of Theorem 4.2.

######  Corollary 4.4.

Let $\Lambda^{\prime}=\\{g_{n}^{\prime}\\}_{n=1}^{\infty}\subset
R^{t}(K,\mu)^{\perp}$ be a dense subset. Then

$\
\mathcal{F}_{0}(\Lambda)\approx\mathcal{F}_{0}(\Lambda^{\prime}),~{}\mathcal{F}_{+}(\Lambda)\approx\mathcal{F}_{+}(\Lambda^{\prime}),~{}\mathcal{F}_{-}(\Lambda)\approx\mathcal{F}_{-}(\Lambda^{\prime}),~{}\gamma-a.a..$

Hence, $\mathcal{F}(\Lambda)$ and $\mathcal{R}(\Lambda)$ are independent of
choices of $\Lambda$ up to a $\gamma$ zero set.

##  5\. Analytic bounded point evaluations for $R^{t}(K,\mu)$

Let $\Gamma_{n}$, $\Gamma$, and $d\mu=hd\mathcal{H}^{1}|_{\Gamma}+d\mu_{s}$ be
as in Lemma 3.2. From Theorem 3.7 and Corollary 4.4, $\mathcal{F}(\Lambda)$
and $\mathcal{R}(\Lambda)$ are independent of choices of $\Gamma_{n}$ and
$\Lambda$ up to a zero set of analytic capacity. This leads the following
definition for $R^{t}(K,\mu).$

######  Definition 5.1.

We define the non-removable boundaries and the removable sets for
$R^{t}(K,\mu)$ as the following:

$\displaystyle\mathcal{F}_{0}=\mathcal{F}_{0}(\Lambda),~{}\mathcal{F}_{+}=\mathcal{F}_{+}(\Lambda),~{}\mathcal{F}_{-}=\mathcal{F}_{-}(\Lambda),\text{
and }\mathcal{F}=\mathcal{F}(\Lambda).$
$\displaystyle\mathcal{R}_{0}=\mathcal{R}_{0}(\Lambda),~{}\mathcal{R}_{1}=\mathcal{R}_{1}(\Lambda),\text{
and }\mathcal{R}=\mathcal{R}(\Lambda).$

Set $\mathcal{E}_{N}=\mathcal{E}_{N}(\Lambda).$

######  Proposition 5.2.

If $S_{\mu}$ on $R^{t}(K,\mu)$ is pure, then
$\text{spt}(\mu)\subset\overline{\mathcal{R}}.$

###### Proof.

If $\mathbb{D}(\lambda_{0},\delta)\subset\mathcal{F}$, from Corollary 4.3, we
see that $\mathcal{C}(g_{j}\mu)(z)=0$ with respect to
${\mathfrak{m}}|_{\mathbb{D}(\lambda_{0},\delta)}-a.a.$ This implies that
$\mu(\mathbb{D}(\lambda_{0},\delta))=0$ by (2.2) since $S_{\mu}$ is pure. ∎

######  Proposition 5.3.

$\partial_{1}K\subset\mathcal{F},~{}\gamma-a.a.$.

###### Proof.

The proposition follows from Theorem 3.7 and the fact that for
$\lambda\in\partial_{1}K$,

$\ \mathbb{D}(\lambda,\delta)\setminus
K\subset\mathbb{D}(\lambda,\delta)\cap\mathcal{E}_{N}.$

∎

The following Lemma is from Lemma B in [2].

######  Lemma 5.4.

There are absolute constants $\epsilon_{1},C_{13}>0$ with the following
property. If $R>0$ and $E\subset\overline{\mathbb{D}(0,R)}$ with
$\gamma(E)<R\epsilon_{1}$, then for $|\lambda|<\frac{R}{2}$ and polynomials
$p$,

$\ |p(\lambda)|\leq\dfrac{C_{13}}{\pi
R^{2}}\int_{\overline{\mathbb{D}(0,R)}\setminus E}|p|\,d{\mathfrak{m}}.$

The theorem below provides an important connection between
$\text{abpe}(R^{t}(K,\mu))$ and $\mathcal{R}.$

######  Theorem 5.5.

The following property holds:

$\displaystyle\
\text{abpe}(R^{t}(K,\mu))\approx\text{int}(K)\cap\mathcal{R},~{}\gamma-a.a..$
(5.1)

More precisely, the following statements are true:

(1) If $\lambda_{0}\in\text{int}(K)$ and there exists $N\geq 1$ such that

$\displaystyle\ \lim_{\delta\rightarrow
0}\dfrac{\gamma(\mathcal{E}_{N}\cap\mathbb{D}(\lambda_{0},\delta))}{\delta}=0,$
(5.2)

then $\lambda_{0}\in\text{abpe}(R^{t}(K,\mu))$.

(2)

$\
\text{abpe}(R^{t}(K,\mu))\subset\text{int}(K)\cap\mathcal{R},~{}\gamma-a.a..$

###### Proof.

(1): If $\lambda_{0}\in\text{int}(K)$ satisfies (5.2), then we choose
$\delta>0$ small enough such that $\mathbb{D}(\lambda_{0},\delta)\subset
int(K)$ and
$\gamma(E:=\mathcal{E}_{N}\cap\mathbb{D}(\lambda_{0},\delta)))\leq\epsilon_{1}\delta$,
where $\epsilon_{1}$ is from Lemma 5.4. Hence, using Lemma 5.4, we conclude

$\ \begin{aligned} \
|r(\lambda)|\leq&\dfrac{C_{13}}{\pi\delta^{2}}\int_{\mathbb{D}(\lambda_{0},\delta)\setminus
E}|r(z)|d{\mathfrak{m}}(z)\\\ \
\leq&\dfrac{NC_{13}}{\pi\delta^{2}}\int_{\mathbb{D}(\lambda_{0},\delta)}|r(z)|\max_{1\leq
j\leq N}|\mathcal{C}(g_{j}\mu)(z)|d{\mathfrak{m}}(z)\\\ \
\leq&\dfrac{NC_{13}}{\pi\delta^{2}}\sum_{j=1}^{N}\int_{\mathbb{D}(\lambda_{0},\delta)}|\mathcal{C}(rg_{j}\mu)(z)|d{\mathfrak{m}}(z)\\\
\
\leq&\dfrac{NC_{13}}{\pi\delta^{2}}\sum_{j=1}^{N}\int\int_{\mathbb{D}(\lambda_{0},\delta)}\left|\dfrac{1}{z-w}\right|d{\mathfrak{m}}(z)|r(w)||g_{j}(w)|d\mu(w)\\\
\
\leq&\dfrac{NC_{14}}{\delta}\sum_{j=1}^{N}\|g_{j}\|_{L^{s}(\mu)}\|r\|_{L^{t}(\mu)}\
\end{aligned}$

for all $\lambda$ in $\mathbb{D}(\lambda_{0},\frac{\delta}{2})$ and all $r\in
Rat(K)$. This implies that $\lambda_{0}\in abpe(R^{t}(K,\mu))$.

(2): Let $E\subset G\cap\mathcal{F}$ be a compact subset with $\gamma(E)>0$,
where $G$ is a connected component of $abpe(R^{t}(K,\mu))$. By Lemma 4.1,
there exists $f\in R^{t,\infty}(K,\mu)$ that is bounded analytic on $E^{c}$
such that $\|f\|_{L^{\infty}(\mu)}\leq
C_{12},~{}f(\infty)=0,~{}f^{\prime}(\infty)=\gamma(E),$ and
$\dfrac{f(z)-f(\lambda)}{z-\lambda}\in R^{t,\infty}(K,\mu)$ for $\lambda\in
E^{c}.$ Let $r_{n}\in Rat(K)$ such that $\|r_{n}-f\|_{L^{t}(\mu)}\rightarrow
0$. Hence, $r_{n}$ uniformly tends to an analytic function $f_{0}$ on compact
subsets of $G$ and $\frac{f-f_{0}(\lambda)}{z-\lambda}\in R^{t}(K,\mu)$ for
$\lambda\in G$. For $\lambda\in G\setminus E$ with $\mu(\\{\lambda\\})=0,$ let
$k_{\lambda}\in L^{s}(\mu)$ such that $r(\lambda)=(r,k_{\lambda})$ for
$r\in\text{Rat}(K).$ Hence, $f_{0}(\lambda)=(f,k_{\lambda}).$ Set
$g=(z-\lambda)\bar{k}_{\lambda}.$ Then $g\perp R^{t}(K,\mu)$ and

$\
0=\int\dfrac{f(z)-f(\lambda)}{z-\lambda}gd\mu=(f-f(\lambda),k_{\lambda})=(f,k_{\lambda})-f(\lambda).$

Therefore, $f(z)=f_{0}(z)$ for $z\in G\setminus E.$ Thus, the function
$f_{0}(z)$ can be analytically extended to $\mathbb{C}_{\infty}$ and
$f_{0}(\infty)=0$. So $f_{0}=0.$ This contradicts to $f^{\prime}(\infty)\neq
0$.

(5.1) now follows from Theorem 3.7. ∎

##  6\. Proof of Theorem A and Theorem B

For a compact subset $E\subset\mathbb{D}(a,\delta)$ and an analytic function
$f$ on $\mathbb{C}_{\infty}\setminus E$ with $f(\infty)=0,$ we consider the
Laurent expansion of $f$ centered at $a$ for
$z\in\mathbb{C}\setminus\mathbb{D}(a,\delta),$

$\ f(z)=\sum_{n=1}^{\infty}\dfrac{c_{n}(f,a)}{(z-a)^{n}}.$

As $c_{1}(f,a)$ does not depend on the choice of $a$, we define
$c_{1}(f)=c_{1}(f,a).$

Let $\varphi$ be a smooth function with compact support. Vitushkin’s
localization operator $T_{\varphi}$ is defined by

$\
(T_{\varphi}f)(\lambda)=\dfrac{1}{\pi}\int\dfrac{f(z)-f(\lambda)}{z-\lambda}\bar{\partial}\varphi(z)d{\mathfrak{m}}(z),$

where $f\in L^{1}_{loc}(\mathbb{C})$. Clearly,
$(T_{\varphi}f)(z)=-\frac{1}{\pi}\mathcal{C}(\varphi\bar{\partial}f{\mathfrak{m}})(z).$
Therefore, by (2.2), $T_{\varphi}f$ is analytic outside of
$\text{supp}(\bar{\partial}f)\cap\text{supp}(\varphi).$ If
$\text{supp}(\varphi)\subset\mathbb{D}(a,\delta),$ then

$\ \|T_{\varphi}f\|_{\infty}\leq
4\|f\|_{\infty}\delta\|\bar{\partial}\varphi\|.$

See [10, VIII.7.1] for the details of $T_{\varphi}.$

Let $\delta>0$. We say that $\\{\varphi_{ij},S_{ij},\delta\\}$ is a smooth
partition of unity subordinated to $\\{2S_{ij}\\},$ if $S_{ij}$ is a square
with vertices
$(i\delta,j\delta),~{}((i+1)\delta,j\delta),~{}(i\delta,(j+1)\delta),$ and
$((i+1)\delta,(j+1)\delta);$ $\varphi_{ij}$ is a $C^{\infty}$ smooth function
supported in $2S_{ij},$ and with values in $[0,1]$; and

$\
\|\bar{\partial}\varphi_{ij}\|\leq\frac{C_{15}}{\delta},~{}\sum\varphi_{ij}=1.$

We shall let $c_{ij}$ denote the center of $S_{ij}$ (See [10, VIII.7] for
details).

Let $f\in C(\mathbb{C}_{\infty})$ (or $f\in L^{\infty}(\mathbb{C})$) with
compact support. Define $f_{ij}=T_{\varphi_{ij}}f,$ then $f_{ij}\neq 0$ for
only finite many $(i,j).$ Clearly, $f(z)=\sum_{f_{ij}\neq 0}f_{ij}(z).$ The
standard Vitushkin approximation scheme requires us to construct functions
$a_{ij}$ such that $f_{ij}-a_{ij}$ has triple zeros at $\infty$, which
requires us to estimate both $c_{1}(a_{ij})$ and $c_{2}(a_{ij},c_{ij})$ (see
[10, section 7 on page 209]). The main idea of P. V. Paramonov [12] is that
one does not actually need to estimate each coefficient
$c_{2}(a_{ij},c_{ij})$. It suffices to estimate the sum of coefficients
$\sum_{j\in J_{il}}c_{2}(a_{ij},c_{ij})$ for a special non-intersecting
partition $\\{J_{il}\\}.$ Let

$\displaystyle\ \alpha_{ij}=\gamma(\mathbb{D}(c_{ij},k_{1}\delta)\cap E),$
(6.1)

where $k_{1}\geq 3$ is a fixed integer. Let
$m_{1}=\min\\{i,j,~{}\mathbb{D}(c_{ij},k_{1}\delta)\cap E\neq\emptyset\\}$ and
$m_{2}=\max\\{i,j,~{}\mathbb{D}(c_{ij},k_{1}\delta)\cap E\neq\emptyset\\}.$ So
$m_{1}$ and $m_{2}$ are finite. Set $min_{i}=\min\\{j:~{}\alpha_{ij}\neq 0\\}$
and $max_{i}=\max\\{j:~{}\alpha_{ij}\neq 0\\}$. Let $J_{i}=\\{j:~{}min_{i}\leq
j\leq max_{i}\\}$.

######  Definition 6.1.

We call a subset $J$ of $J_{i}$ a complete group if $J=\\{j:~{}j_{1}+1\leq
j\leq j_{1}+s_{1}+s_{2}+s_{3}\\},$ where $s_{2}$ is an absolute constant that
will be chosen in Lemma 6.6, and $s_{1}$ and $s_{3}$ are chosen so that

$\
\delta\leq\sum_{j=j_{1}+1}^{j_{1}+s_{1}}\alpha_{ij}<\delta+k_{1}\delta\text{
and
}\delta\leq\sum_{j=j_{1}+s_{1}+s_{2}+1}^{j_{1}+s_{1}+s_{2}+s_{3}}\alpha_{ij}<\delta+k_{1}\delta.$

We now present a detailed description of the procedure of partitioning $J_{i}$
into groups. We split each $J_{i}$ into (finitely many) non-intersecting
groups $J_{il}$, $l=1,...,l_{i}$, as follows. Starting from the lowest index
$min_{i}$ in $J_{i}$ we include in $J_{i1}$ (going upwards and without jumps
in $J_{i}$) all indices until we have collected a minimal (with respect to the
number of elements) complete group $J_{i1}$. Then we repeat this procedure for
$J_{i}\setminus J_{i1}$, and so on. After we have constructed all the complete
groups $J_{i1},...,J_{il_{i}-1}$ in this way (there may be none), then what
remains is the last portion $J_{il_{i}}=J_{i}\setminus(J_{i1}\cup...\cup
J_{il_{i}-1})$ of indices in $J_{i}$, which includes no complete groups. We
call this portion $J_{il_{i}}$ an incomplete group of indices (clearly, there
is at most one incomplete group for each $i$).

######  Definition 6.2.

Let $\delta>0,$ $\alpha_{ij},$ and $E$ be defined as in (6.1). Let $P_{1}$ be
the set of all complete groups $J_{il}$ for $1\leq l\leq l_{i}-1$ and
$m_{1}\leq i\leq m_{2}.$ Let $P_{2}$ be the set of all incomplete groups
$J_{il_{i}}$ for $m_{1}\leq i\leq m_{2}.$ Set $P=P_{1}\cup P_{2}.$

For a group $J$ (complete or incomplete) with row index $i,$ let

$\displaystyle\ J^{\prime}(z)=\\{j\in J:~{}\alpha_{ij}>0\text{ and
}|z-c_{ij}|>3k_{1}\delta\\},$ (6.2)

and

$\displaystyle\ L_{J}^{\prime}(z)=\sum_{j\in
J^{\prime}(z)}\left(\dfrac{\delta\alpha_{ij}}{|z-c_{ij}|^{2}}+\dfrac{\delta^{3}}{|z-c_{ij}|^{3}}\right).$
(6.3)

Define $L_{J}(z)=L_{J}^{\prime}(z)$ if $\\{j\in
J:~{}\alpha_{ij}>0\\}=J^{\prime}(z)$, otherwise,
$L_{J}(z)=1+L_{J}^{\prime}(z)$. For $g_{ij}$ that is bounded and analytic on
$\mathbb{C}\setminus E_{ij},$ where $E_{ij}$ is a compact subset of
$\mathbb{D}(c_{ij},k_{1}\delta),$ define

$\displaystyle\ g_{J}=\sum_{j\in J}g_{ij},~{}c_{1}(g_{J})=\sum_{j\in
J}c_{1}(g_{ij}),~{}c_{2}(g_{J})=\sum_{j\in J}c_{2}(g_{ij},c_{ij}).$ (6.4)

######  Definition 6.3.

A sequence of functions $\\{g_{ij}\\}$ is applicable to $P$ and $\mu\in
M_{0}^{+}(\mathbb{C})$ if the following assumptions hold, for some absolute
constant $C_{16}$:

(1) $g_{ij}$ is bounded and analytic on
$\mathbb{C}_{\infty}\setminus\overline{\mathbb{D}(c_{ij},k_{1}\delta)};$

(2) $g_{ij}(\infty)=c_{1}(g_{ij})=0;$

(3) $\|g_{ij}\|_{L^{\infty}(\mu)}\leq C_{16}$;

(4) $|c_{n}(g_{ij},c_{ij})|\leq C_{16}\delta^{n-1}\alpha_{ij}$ for $n\geq 1.$

The following lemma is straightforward.

######  Lemma 6.4.

If $f(z)$ is bounded analytic on
$\mathbb{C}_{\infty}\setminus\overline{\mathbb{D}(a,\delta)}$ with
$f(\infty)=0,$ $\|f\|\leq 1,$ $E\subset\overline{\mathbb{D}(a,\delta)},$ and

$\ |c_{n}(f,a)|\leq C_{17}n\delta^{n-1}\gamma(E).$

Then for $|z-a|>2\delta,$

$\
\left|f(z)-\dfrac{c_{1}(f)}{z-a}-\dfrac{c_{2}(f,a)}{(z-a)^{2}}\right|\leq\dfrac{C_{18}\gamma(E)\delta^{2}}{|z-a|^{3}}.$

Therefore, by Lemma 6.4, if $\\{g_{ij}\\}$ is applicable to $P,$ then for
$|z-c_{ij}|>3k_{1}\delta,$

$\displaystyle\
|g_{ij}(z)|\leq\dfrac{C_{19}\delta\alpha_{ij}}{|z-c_{ij}|^{2}}+\dfrac{C_{19}\delta^{3}}{|z-c_{ij}|^{3}}.$
(6.5)

The following lemma follows easily from Definitions 6.2 and (6.5).

######  Lemma 6.5.

Let $\\{g_{ij}\\}$ be applicable to $P$ and $\mu\in M_{0}^{+}(\mathbb{C})$. If
$J$ is a group (complete or incomplete) with row index $i,$ then

$\displaystyle|g_{J}(z)|\leq
C_{20}L_{J}(z),~{}\|g_{J}\|(\|g_{J}\|_{L^{\infty}(\mu)})\leq C_{20},$
$\displaystyle c_{1}(g_{J})=0,~{}|c_{2}(g_{J})|\leq C_{20}\delta^{2}.$

The following key lemma is due to [12, Lemma 2.7], where $s_{2}$ is also
selected.

######  Lemma 6.6.

Let $\\{g_{ij}\\}$ be applicable to $P$ and $\mu\in M_{0}^{+}(\mathbb{C}).$
Suppose that there exists $h_{ij}\in L^{\infty}(\mu)$ that is bounded and
analytic on
$\mathbb{C}_{\infty}\setminus\overline{\mathbb{D}(c_{ij},k_{1}\delta)}$ and
satisfy

$\displaystyle h_{ij}(\infty)=0,~{}\|h_{ij}\|\leq
C_{21},~{}c_{1}(h_{ij})=\alpha_{ij},$ $\displaystyle|c_{n}(h_{ij},c_{ij})|\leq
C_{21}\delta^{n-1}\alpha_{ij}\text{ for }n\geq 1.$

Then for each complete group $J_{il}\in P_{1}$, there exists a function
$h_{J_{il}}$ that is a linear combination of $h_{ij}$ for $j\in J_{il}^{d}\cup
J_{il}^{u},$ where $J_{il}^{d}=(j_{1}+1,...,j_{1}+s_{1})$ and
$J_{il}^{u}=(j_{1}+s_{1}+s_{2}+1,...,j_{1}+s_{1}+s_{2}+s_{3})$, such that for
all $z\in\mathbb{C},$

$\displaystyle\begin{aligned} \ &|h_{J_{il}}(z)|\leq
C_{22}L_{J_{il}}(z),~{}\|h_{J_{il}}\|\leq C_{22},\\\ \
&c_{1}(h_{J_{il}})=0,~{}c_{2}(h_{J_{il}})=c_{2}(g_{J_{il}}).\end{aligned}$
(6.6)

The following lemma is also from [12] with slight modifications.

######  Lemma 6.7.

Let $\\{g_{ij}\\}$ be applicable to $P$ and $\mu\in M_{0}^{+}(\mathbb{C}).$
Let $\\{h_{ij}\\}$ satisfy the assumption in Lemma 6.6. Suppose that for each
$J_{il}\in P_{1},$ $h_{J_{il}}$ is constructed as in Lemma 6.6. Set
$\Psi_{J_{il}}(z)=g_{J_{il}}(z)-h_{J_{il}}(z).$ Then

$\displaystyle\ \sum_{J_{il}\in P_{1}}|\Psi_{J_{il}}(z)|+\sum_{J_{il_{i}}\in
P_{2}}|g_{J_{il_{i}}}(z)|\leq
C_{23}\min\left(1,\dfrac{\delta}{dist(z,E)}\right)^{\frac{1}{5}}.$ (6.7)

###### Proof.

We just need to modify the proof in [12] slightly. We fix $z\in\mathbb{C}.$
Let $N$ be the largest integer that is smaller that
$\frac{dist(z,E)}{2\delta}.$ If $N^{\frac{4}{5}}\leq 3k_{1}+4s_{2},$ then
(6.7) is bounded by an absolute constant $C_{23}$ as showed in [12].

Assume that $N^{\frac{4}{5}}>3k_{1}+4s_{2}.$ We make the following
modifications
(1) For $J\in P$ with row index $i,$ we add $\alpha_{ij}>0$ to the definitions
of $J^{\prime}(z)$ in (6.2) and $L_{J}(z)$ in the line after in (6.3)
($I^{\prime}(z)$ in [12]). By (6.3), $L_{J}(z)=L_{J}^{\prime}(z).$ Let $j_{J}$
be the largest positive integer that is smaller that $\min_{j\in
J^{\prime}(z)}|j-\frac{Im(z)}{\delta}|.$ Then $j_{J}>N$ and
$L_{J}(z)\leq\frac{C_{25}}{(i-\frac{Re(z)}{\delta})^{2}+j_{J}^{2}}.$
(2) For $P_{2},$ we have

$\ \sum_{J_{il_{i}}\in P_{2}}|g_{J_{il_{i}}}(z)|\leq
C_{26}\sum_{i}\dfrac{1}{(i-\frac{Re(z)}{\delta})^{2}+j_{J_{il_{i}}}^{2}}\leq\dfrac{C_{27}}{N}.$

(3) For $P_{1}$ with $|i-\frac{Re(z)}{\delta}|\leq N^{\frac{4}{5}},$ we have

$\ \sum_{\underset{J_{il}\in P_{1}}{|i-\frac{Re(z)}{\delta}|\leq
N^{\frac{4}{5}}}}|\Psi_{J_{il}}(z)|\leq
C_{28}\sum_{|i-\frac{Re(z)}{\delta}|\leq
N^{\frac{4}{5}}}\sum_{j>N}\dfrac{1}{(i-\frac{Re(z)}{\delta})^{2}+j^{2}}\leq\dfrac{C_{29}}{N^{\frac{1}{5}}}.$

(4) For $P_{1}$ with $|i-\frac{Re(z)}{\delta}|>N^{\frac{4}{5}},$ we modify the
case $|m|>4k_{2}$ in [12] (replace $4k_{2}$ by $N^{\frac{4}{5}}$) and get

$\ \sum_{\underset{J_{il}\in
P_{1}}{|i-\frac{Re(z)}{\delta}|>N^{\frac{4}{5}}}}|\Psi_{J_{il}}(z)|\leq
C_{30}\sum_{m\geq
N^{\frac{4}{5}}}\dfrac{1}{m^{\frac{5}{4}}}\leq\dfrac{C_{31}}{N^{\frac{1}{5}}}.$

The lemma is proved. ∎

We now start our proofs for Theorem A and Theorem B.

######  Lemma 6.8.

If $U$ is an open subset of $\text{abpe}(R^{t}(K,\mu))$ satisfying $\partial
U\subset\mathcal{F},~{}\gamma-a.a.,$ then, for $f\in H^{\infty}(U)$ with
$\|f\|_{U}\leq 1$ and $f(z)=0$ for $z\in\mathbb{C}\setminus U,$ there exists
$\tilde{f}\in R^{t,\infty}(K,\mu)$ such that
$\tilde{f}(z)=0,~{}\mu|_{\overline{U}^{c}}-a.a.,$
$f(z)\mathcal{C}(g\mu)(z)=\mathcal{C}(\tilde{f}g\mu)(z),~{}{\mathfrak{m}}-a.a.$
for $g\perp R^{t}(K,\mu),$ and $\|\tilde{f}\|_{L^{\infty}(\mu)}\leq C_{32}.$

###### Proof.

Let $k_{1}=3$ and $E=\partial U.$ Let $\alpha_{ij}$ be as in (6.1). Let $P,$
$P_{1},$ and $P_{2}$ be as in Definition 6.2. We fix
$z\in\mathbb{C}\setminus\partial U.$ Then

$\displaystyle\
f=\sum_{2S_{ij}\cap\partial\Omega\neq\emptyset}(f_{ij}:=T_{\varphi_{ij}}f).$
(6.8)

The function $f_{ij}$ is analytic on
$\mathbb{C}_{\infty}\setminus(\overline{\mathbb{D}(c_{ij},\sqrt{2}\delta)}\cap\partial
U)$ and therefore,

$\displaystyle\begin{aligned} \
|c_{n}(f_{ij},c_{ij})|=&\dfrac{1}{\pi}\left|\int
f(z)(z-c_{ij})^{n-1}\bar{\partial}\varphi_{ij}(z)d{\mathfrak{m}}(z)\right|\\\
\
\leq&\|T_{\varphi_{ij}}((z-c_{ij})^{n-1}f)\|\gamma(\overline{\mathbb{D}(c_{ij},\sqrt{2}\delta)}\cap\partial
U).\\\ \ \leq&C_{33}\delta^{n-1}\alpha_{ij}.\end{aligned}$ (6.9)

Let $g_{ij}^{0}$ be as in Lemma 4.1 for a compact subset
$E_{1}\subset\mathbb{D}(c_{ij},k_{1}\delta)\cap\partial U\subset\mathcal{F}$
with $\gamma(E_{1})\geq\frac{1}{2}\alpha_{ij}$ when $\alpha_{ij}>0.$ Let
$g_{ij}^{1}=-\frac{c_{1}(f_{ij})}{\gamma(E_{1})}g_{ij}^{0},$
$g_{ij}=f_{ij}-g_{ij}^{1},$ and
$h_{ij}=-\frac{\alpha_{ij}}{\gamma(E_{1})}g_{ij}^{0}.$ Clearly, by (6.9), we
see that $\\{g_{ij}\\}$ is applicable to $P$ (see Definition 6.3) and
$\\{h_{ij}\\}$ satisfies the assumptions in Lemma 6.6. The function
$g_{J_{il}}$ defined in (6.4) satisfies the properties in Lemma 6.5. Let
$h_{J_{il}}$ be as in Lemma 6.6.

We rewrite (6.8) as the following

$\ f=\sum_{J_{il}\in P_{1}}(g_{J_{il}}-h_{J_{il}})+\sum_{J_{il_{i}}\in
P_{2}}g_{J_{il_{i}}}+f_{\delta}$

where

$\displaystyle\ f_{\delta}=\sum_{J_{il}\in P}\sum_{j\in
J_{il}}g_{ij}^{0}+\sum_{J_{il}\in P_{1}}h_{J_{il}}\in R^{t,\infty}(K,\mu).$
(6.10)

From (6.7), we get

$\ |f(z)-f_{\delta}(z)|\leq C_{23}\min\left(1,\dfrac{\delta}{dist(z,\partial
U)}\right)^{\frac{1}{5}}.$

Therefore, $f_{\delta}(z)\rightarrow f(z)$ uniformly on compact subsets of
$\mathbb{C}_{\infty}\setminus\partial U$ and
$\|f_{\delta}\|_{L^{\infty}(\mu)}\leq C_{32}.$There exists a subsequence
$\\{f_{\delta_{n}}\\}$ such that $f_{\delta_{n}}(z)\rightarrow\tilde{f}(z)$ in
$L^{\infty}(\mu)$ weak-star topology. Hence,
$\|\tilde{f}\|_{L^{\infty}(\mu)}\leq C_{32},$ $\tilde{f}\in
R^{t,\infty}(K,\mu),$ and
$\mathcal{C}(f_{\delta_{n}}g\mu)(z)\rightarrow\mathcal{C}(\tilde{f}g\mu)(z),~{}{\mathfrak{m}}-a.a..$
From (4.1) and Corollary 4.3, we infer that

$\
f_{\delta_{n}}(z)\mathcal{C}(g\mu)(z)=\mathcal{C}(f_{\delta_{n}}g\mu)(z),~{}{\mathfrak{m}}-a.a..$

Thus,
$f(z)\mathcal{C}(g\mu)(z)=\mathcal{C}(\tilde{f}g\mu)(z),~{}{\mathfrak{m}}-a.a.$
for $g\perp R^{t}(K,\mu).$ ∎

###### Proof.

(Theorem A) There exists a partition $\\{\Delta_{00},\Delta_{01}\\}$ of
$\text{spt}\mu$ such that

$\ R^{t}(K,\mu)=L^{t}(\mu|_{\Delta_{00}})\oplus R^{t}(K,\mu|_{\Delta_{01}})$

and $S_{\mu|_{\Delta_{01}}}$ is pure. Let $\mathcal{F}_{\Delta_{01}}$ be the
non-removable boundary for $R^{t}(K,\mu|_{\Delta_{01}}).$ If $g\perp
R^{t}(K,\mu),$ then $g(z)=0,~{}\mu|_{\Delta_{00}}-a.a..$ Hence, by Theorem
3.7, we see that $\mathcal{F}_{\Delta_{01}}\approx\mathcal{F},~{}\gamma-a.a..$

Let $\mathbb{D}(\lambda_{0},\delta)\subset\text{abpe}(R^{t}(K,\mu))$ such that
for $\lambda\in\mathbb{D}(\lambda_{0},\delta)$ and $r\in\text{Rat}(K),$
$|r(\lambda)|\leq M\|r\|_{L^{t}(\mu)}$ for some $M>0$ and
$r(\lambda)=(r,k_{\lambda}),$ where $k_{\lambda}\in L^{s}(\mu)$ and
$\|k_{\lambda}\|_{L^{s}(\mu)}\leq M.$ Because
$(z-\lambda)\bar{k}_{\lambda}\perp R^{t}(K,\mu),$ we get
$k_{\lambda}(z)=0,~{}\mu|_{\Delta_{00}}-a.a.$ if $\mu(\\{\lambda\\})=0.$ Let
$\\{\lambda_{n}\\}$ be the set of atoms for $\mu.$ Then $|r(\lambda)|\leq
M\|r\|_{L^{t}(\mu|_{\Delta_{01}})}$ for
$\lambda\in\mathbb{D}(\lambda_{0},\delta)\setminus\\{\lambda_{n}\\}.$ Now for
$|\lambda-\lambda_{0}|<\frac{\delta}{2},$ we have

$\
|r(\lambda)|\leq\dfrac{C_{35}}{\pi\delta^{2}}\int_{\mathbb{D}(\lambda_{0},\delta)\setminus\\{\lambda_{n}\\}}|r(z)|dm\leq
C_{35}M\|r\|_{L^{t}(\mu|_{\Delta_{01}})}.$

Thus, $\lambda_{0}\in\text{abpe}(R^{t}(K,\mu|_{\Delta_{01}}))$ which implies
$\text{abpe}(R^{t}(K,\mu|_{\Delta_{01}}))=\text{abpe}(R^{t}(K,\mu).$

Therefore, the assumption of Theorem A holds for $R^{t}(K,\mu|_{\Delta_{01}})$
and we may assume that $S_{\mu}$ is pure in the following proof.

From (5.1), we see that $\partial
U_{i}\cap\text{int}(K)\subset\mathcal{F},~{}\gamma-a.a..$ Hence, the
assumption $\partial U_{i}\cap\partial K\subset\mathcal{F},~{}\gamma-a.a.$
implies $\partial U_{i}\subset\mathcal{F},~{}\gamma-a.a..$

Let $f_{i}=\chi_{U_{i}}.$ By Lemma 6.8, there exists $\tilde{f}_{i}\in
R^{t,\infty}(K,\mu)$ such that
$\tilde{f}_{i}(z)=0,~{}z\in\overline{U}_{i}^{c}$ and
$f_{i}(z){\mathcal{C}}(g\mu)(z)={\mathcal{C}}(\tilde{f}_{i}g\mu)(z),~{}{\mathfrak{m}}-a.a.$
for $g\perp R^{t}(K,\mu).$ Thus, for $g\perp R^{t}(K,\mu),$

$\
{\mathcal{C}}(\tilde{f}_{i}g\mu)(z)=f_{i}(z)(f_{i}(z){\mathcal{C}}(g\mu)(z))=f_{i}(z){\mathcal{C}}(\tilde{f}_{i}g\mu)(z)={\mathcal{C}}(\tilde{f}_{i}^{2}g\mu)(z),~{}{\mathfrak{m}}-a.a.$

since $\tilde{f}_{i}g\perp R^{t}(K,\mu).$ So, since $S_{\mu}$ is pure, we have

$\ \tilde{f}_{i}(z)=\tilde{f}_{i}^{2}(z),~{}\mu-a.a..$

Hence, there exists a Borel subset $\Delta_{i}$ with
$U_{i}\subset\Delta_{i}\subset\overline{U}_{i}$ such that
$\tilde{f}_{i}=\chi_{\Delta_{i}}.$ So

$\displaystyle\
{\mathcal{C}}(\chi_{\Delta_{i}}g\mu)(z)=\chi_{U_{i}}(z){\mathcal{C}}(g\mu)(z),~{}{\mathfrak{m}}-a.a.,\text{
for }i\geq 1.$ (6.11)

Therefore,

$\
{\mathcal{C}}(\chi_{\Delta_{i}}\chi_{\Delta_{m}}g\mu)(z)=\chi_{U_{i}}(z)\chi_{U_{m}}(z){\mathcal{C}}(g\mu)(z)=0,~{}{\mathfrak{m}}-a.a.,$

for $i,m\geq 1$ and $i\neq m,$ which implies
$\Delta_{i}\cap\Delta_{m}=\emptyset,~{}\mu-a.a.$ as $S_{\mu}$ is pure. Set
$\Delta_{0}=K\setminus\cup_{i=1}^{\infty}\Delta_{i}.$ Then

$\
R^{t}(K,\mu)=R^{t}(K,\mu|_{\Delta_{0}})\oplus\bigoplus_{i=1}^{\infty}R^{t}(K,\mu|_{\Delta_{i}}).$

For $i\geq 1,$ it is clear that
$R^{t}(K,\mu|_{\Delta_{i}})=R^{t}(\overline{U_{i}},\mu|_{\Delta_{i}})$ and so
(c) holds.

Let $K_{0}$ be the spectrum of $S_{\mu|_{\Delta_{0}}}.$ Then
$R^{t}(K_{0},\mu|_{\Delta_{0}})=R^{t}(K,\mu|_{\Delta_{0}})$ and for $i\geq 1,$
$K_{0}\cap U_{i}=\emptyset$ since, by (6.11),

$\
{\mathcal{C}}(\chi_{\Delta_{0}}g\mu)(z)={\mathcal{C}}(g\mu)(z)-\sum_{m=1}^{\infty}{\mathcal{C}}(\chi_{\Delta_{m}}g\mu)(z)=0,~{}{\mathfrak{m}}|_{U_{i}}-a.a..$

Therefore, $\text{abpe}(R^{t}(K_{0},\mu|_{\Delta_{0}}))=\emptyset.$ This
proves (a).

For $f\in R^{t,\infty}(K,\mu|_{\Delta_{i}})$ and $\lambda\in U_{i},$ we have
$\frac{f(z)-\rho_{i}(f)(\lambda)}{z-\lambda}\in
R^{t,\infty}(K,\mu|_{\Delta_{i}}).$ Hence, by Corollary 4.3,
${\mathcal{C}}(fg\mu)(\lambda)=\rho_{i}(f)(\lambda){\mathcal{C}}(g\mu)((\lambda)),~{}{\mathfrak{m}}-a.a.$
for $g\perp R^{t}(K,\mu|_{\Delta_{i}}).$ As $S_{\mu_{\Delta_{i}}}$ is pure, we
conclude that $\rho_{i}$ is injective and

$\ \rho_{i}(f_{1}f_{2})=\rho_{i}(f_{1})\rho_{i}(f_{2}),~{}f_{1},f_{2}\in
R^{t,\infty}(K,\mu|_{\Delta_{i}}).$

(b): If $B\subset\Delta_{i}$ and $\chi_{B}\in
R^{t,\infty}(K,\mu|_{\Delta_{i}})$ is a non-zero characteristic function. Then
$(\rho_{i}(\chi_{B}))^{2}=\rho_{i}(\chi_{B}),$ which implies
$\rho_{i}(\chi_{B})=\rho_{i}(\chi_{\Delta_{i}})=1$ since $U_{i}$ is connected.
Hence, $\chi_{B}=\chi_{\Delta_{i}}$ as $\rho_{i}$ is injective. So
$S_{\mu|_{\Delta_{i}}}$ is irreducible.

(d): For $f\in R^{t,\infty}(K,\mu|_{\Delta_{i}})$ and $\lambda\in U_{i},$
there is a constant $C_{\lambda}>0$ such that

$\ |\rho_{i}(f)(\lambda)|\leq
C_{\lambda}^{\frac{1}{n}}\|f^{n}\|_{L^{t}(\mu|_{\Delta_{i}})}^{\frac{1}{n}}.$

Taking $n\rightarrow\infty,$ we get
$\|\rho_{i}(f)\|_{U_{i}}\leq\|f\|_{L^{\infty}(\mu|_{\Delta_{i}})}.$ From Lemma
6.8, we see $\rho_{i}$ is surjective. Moreover,
$\|f^{n}\|_{L^{\infty}(\mu|_{\Delta_{i}})}^{\frac{1}{n}}\leq
C_{32}^{\frac{1}{n}}\|\rho_{i}(f^{n})\|_{U_{i}}^{\frac{1}{n}}.$ Letting
$n\rightarrow\infty,$ we get
$\|f\|_{L^{\infty}(\mu|_{\Delta_{i}})}\leq\|\rho_{i}(f)\|_{U_{i}}.$ Thus,

$\ \|\rho_{i}(f)\|_{U_{i}}=\|f\|_{L^{\infty}(\mu|_{\Delta_{i}})},~{}f\in\in
R^{t,\infty}(K_{i},\mu|_{\Delta_{i}}).$

Therefore, $\rho_{i}$ is a bijective isomorphism between two Banach algebras
$R^{t,\infty}(K_{i},\mu|_{\Delta_{i}})$ and $H^{\infty}(U_{i})$. Clearly
$\rho_{i}$ is also weak-star sequentially continuous, so an application of the
Krein-Smulian Theorem shows that $\rho_{i}$ is a weak-star homeomorphism.

∎

###### Proof.

(Theorem B) Let $\text{abpe}(R^{t}(K,\mu))=\cup_{i=1}^{\infty}U_{i},$ where
$U_{i}$ is a connected component. By the assumption and Proposition 5.3,

$\ \partial U_{i}\cap\partial
K\subset\partial_{1}K\subset\mathcal{F},~{}\gamma-a.a..$

Therefore, by Theorem A, we see that the decomposition in Theorem A holds. We
only need to show that

$\ R^{t}(K,\mu|_{\Delta_{0}})=L^{t}(\mu|_{\Delta_{0}}).$

In fact, from (5.1), if $\mathcal{F}_{\Delta_{0}}$ is the non-removable
boundary for $R^{t}(K,\mu|_{\Delta_{0}}),$ then

$\
\text{int}(K)\setminus\mathcal{F}_{\Delta_{0}}\approx\text{abpe}(R^{t}(K,\mu|_{\Delta_{0}}))=\emptyset,~{}\gamma-a.a..$

Hence, $\mathcal{F}_{\Delta_{0}}=\mathbb{C},~{}\gamma-a.a.$ since $\partial
K=\partial_{1}K\subset\mathcal{F}_{\Delta_{0}},~{}\gamma-a.a.$ by Proposition
5.3. The proof follows from Corollary 4.3. ∎

Acknowledgments. The authors would like to thank Professor John McCarthy for
carefully reading through the manuscript and providing many useful comments.

## References

  * [1] J. R. Akeroyd, J B. Conway, and L.Yang, On non-tangential limits and shift invariant subspaces, Integral Equation and Operator Theory 91 (2019), no,1, Paper No. 2, 18 pp.
  * [2] A. Aleman, S. Richter, and C. Sundberg, Nontangential limits in $P^{t}(\mu)$-spaces and the index of invariant subspaces, Ann. of Math. 169 (2009), no. 2, 449–490.
  * [3] J. E. Brennan, The structure of certain spaces of analytic functions, Comput. Methods Funct. theory 8 (2008), no. 2, 625–640.
  * [4] L. Carleson, Selected problems on exceptional sets, Van Nostrand. Services, Princeton NJ, 1967
  * [5] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Chauchy définit un opératour borné sur $L^{2}$ pour les courbes lipschitzennes, Ann. of Math. 116 (1982), no. 2, 361–387.
  * [6] J. B. Conway, The theory of subnormal operators, Mathematical Survey and Monographs 36, 1991.
  * [7] J. B. Conway and N. Elias, Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis 117 (1993), 1–24.
  * [8] J B. Conway and L.Yang, Mean Rational Approximation for Some Compact Planar Subsets, https://arxiv.org/pdf/2212.05392.pdf.
  * [9] G. David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479.
  * [10] T. W. Gamelin, Uniform algebras, American Mathematical Society, Rhode Island, 1969.
  * [11] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Volume 44 of Cambridge Studies in Mathematics, Cambridge University Press, Cambridge, 1995
  * [12] P. V. Paramonov, Some new criteria for uniform approximability of functions by rational fractions, Sbornik: Mathematics, 186 (1995), no. 9, 1325–1340
  * [13] J. E. Thomson, Approximation in the mean by polynomials, Ann. of Math. 133 (1991), no. 3, 477–507.
  * [14] X. Tolsa, On the analytic capacity $\gamma+$, Indiana Univ. Math.J. 51 (2002), no. 2, 317–343.
  * [15] X. Tolsa, Painleves problem and the semiadditivity of analytic capacity, Acta Math. 51 (2003), no. 1, 105–149.
  * [16] X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous calderon-zygmund theory, Birkhauser/Springer, Cham, 2014.