Positive harmonic boundary measures and compact normalized families

Nonnegative integrands have nonnegative integrals against finite nonnegative measures. Let $\rho$ be a finite measure on $\mathbb{T}$ and let $h:\mathbb{T}\to \mathbb{R}$ be bounded Borel with $h\ge 0$, $M:={\sup }_{\mathbb{T}}h<\mathrm{\infty }$. Because $\rho$ is countably additive with values in $[0,\mathrm{\infty })$ it is a finite signed measure, and for every countable Borel partition of a Borel set $E$ one has ${\sum }_j\mathrm{\mid }\rho (E_j)\mathrm{\mid }={\sum }_j\rho (E_j)=\rho (E)$, so $\mathrm{\mid }\rho \mathrm{\mid }(E)=\rho (E)$ and $h\in L^1(\rho )=L^1(\mathrm{\mid }\rho \mathrm{\mid })$ since ${\int }_{\mathbb{T}}\mathrm{\mid }h\mathrm{\mid }d\rho \le M\rho (\mathbb{T})<\mathrm{\infty }$. Let $s_n$ be increasing nonnegative simple functions with $s_n\uparrow h$ and $0\le s_n\le h$. Then $\mathrm{\mid }h-s_n\mathrm{\mid }\le M$ with the constant $M$ integrable for the finite measure $\rho$, so ${\int }_{\mathbb{T}}\mathrm{\mid }h-s_n\mathrm{\mid }d\rho \to 0$, and $(s_n)$ is an admissible approximating sequence in the definition of ${\int }_{\mathbb{T}}hd\rho$; hence ${\int }_{\mathbb{T}}hd\rho ={\lim }_n{\int }_{\mathbb{T}}{s}_{n}d\rho$. Writing $s_n={\sum }_j{c}_j{\mathbf{1}}_{E_j}$ in canonical form, all $c_j>0$ and all $\rho (E_j)\ge 0$, so ${\int }_{\mathbb{T}}{s}_{n}d\rho ={\sum }_j{c}_j\rho (E_j)\ge 0$ for every $n$ and therefore ${\int }_{\mathbb{T}}hd\rho \ge 0$.

The nonnegative harmonic $u$ lies in $h^1(\mathbb{D})$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}=u(0)$. For $0\le r<1$ the function $u_r(\zeta )=u(r\zeta )$ is continuous on $\mathbb{T}$ and $u_r\ge 0$. If $r=0$ then ${\int }_{\mathbb{T}}{u}_{0}dm=u(0)$. If $r>0$, then ${\int }_{\mathbb{T}}{u}_{r}dm=\frac{1}{2\pi }{\int }_0^{2\pi }u(r{e}^{i\theta })d\theta =u(0)$, the first equality by the torus identification combined with the change of variables $t\mapsto 2\pi t$ and the second by the circle mean-value property applied to the harmonic $u$ on the disc $\overline{D(0,r)}\subseteq \mathbb{D}$. Hence $\mathrm{\parallel }{u}_r{\mathrm{\parallel }}_{L^1(\mathbb{T},m)}={\int }_{\mathbb{T}}\mathrm{\mid }{u}_r\mathrm{\mid }dm={\int }_{\mathbb{T}}{u}_{r}dm=u(0)$ for every radius, so ${\sup }_{0\le r<1}\mathrm{\parallel }{u}_r{\mathrm{\parallel }}_1=u(0)<\mathrm{\infty }$: $u$ is complex harmonic, its imaginary part being the harmonic zero function, and therefore $u\in h^1(\mathbb{D})$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}=u(0)$.

Kernel Lipschitz bound on compacta. Let $K\subseteq \mathbb{D}$ be compact. For $K=\mathrm{\varnothing }$ the estimate is vacuous; assume $K\mathrm{\ne }\mathrm{\varnothing }$. The open discs $U_n=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1-1\mathrm{/}n\}$, $n\ge 2$, cover $\mathbb{D}$ and hence $K$, so [L15] gives $N\ge 2$ with $K\subseteq U_N$; thus $\mathrm{\mid }\phi (\zeta )-z\mathrm{\mid }\ge 1-\mathrm{\mid }z\mathrm{\mid }>1\mathrm{/}N$ for all $z\in K$ and $\zeta \in \mathbb{T}$. Fix $z,z^{\mathrm{\prime }}\in K$ and $w:=\phi (\zeta )$ for $\zeta \in \mathbb{T}$, and set $a:=\mathrm{\mid }w-z{\mathrm{\mid }}^2$, $b:=\mathrm{\mid }w-z^{\mathrm{\prime }}{\mathrm{\mid }}^2$. Then $P(z,\zeta )-P(z^{\mathrm{\prime }},\zeta )=[(1-\mathrm{\mid }z{\mathrm{\mid }}^2)b-(1-\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2)a]\mathrm{/}(ab)$ and the numerator equals $(1-\mathrm{\mid }z{\mathrm{\mid }}^2)(b-a)+(\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2-\mathrm{\mid }z{\mathrm{\mid }}^2)a$, where $\mathrm{\mid }b-a\mathrm{\mid }\le 4\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }$ and $\mathrm{\mid }(\mathrm{\mid }{z}^{\mathrm{\prime }}{\mathrm{\mid }}^2-\mathrm{\mid }z{\mathrm{\mid }}^2)a\mathrm{\mid }\le 2\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }\cdot 4$; hence the numerator has modulus at most $12\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }$ while $ab\ge N^{-4}$. Therefore $\mathrm{\mid }P(z,\zeta )-P(z^{\mathrm{\prime }},\zeta )\mathrm{\mid }\le 12N^4\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }$ for every $\zeta \in \mathbb{T}$.

$C(\mathbb{T},\mathbb{C})$ is separable. By [L12] the torus $\mathbb{T}$ is homeomorphic to the compact metric space $S^1$, hence is itself a compact metric space; by [L13], and countable choice is available by [L11], there is a sequence $(f_j)$ in $C(\mathbb{T},\mathbb{R})$ dense for the supremum norm. The family $\{f_j+i{f}_k:j,k\ge 1\}$ is at most countable by [L13] and is dense in $C(\mathbb{T},\mathbb{C})$: for $g=v+iw$ and $\epsilon >0$ choose $j,k$ with $\mathrm{\parallel }v-f_j{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon \mathrm{/}2$ and $\mathrm{\parallel }w-f_k{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon \mathrm{/}2$, so that $\mathrm{\parallel }g-(f_j+i{f}_k){\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$. Hence $C(\mathbb{T},\mathbb{C})$ has an at most countable dense subset, that is, it is separable.

Representation of $u$. By [L1] applied to $u\in h^1(\mathbb{D})$ there is a unique finite regular complex Borel measure $\mu$ on $\mathbb{T}$ with $u=P[\mu ]$, $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})=\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}=u(0)$, and against $C(\mathbb{T})$ as $r\uparrow 1$.

Converse direction (b). Let $\rho$ be a finite nonnegative regular Borel measure on $\mathbb{T}$. It is a finite regular complex Borel measure, so by the converse clause of [L1] the function $P[\rho ]$ lies in $h^1(\mathbb{D})$ and is harmonic. For $z\in \mathbb{D}$ the function $P(z,\cdot )$ is continuous by [L5], hence bounded Borel, and positive, so step 1.1 with the finite measure $\rho$ and $h=P(z,\cdot )$ gives $P[\rho ](z)={\int }_{\mathbb{T}}P(z,\zeta )d\rho (\zeta )\ge 0$. Moreover $P[\rho ](0)={\int }_{\mathbb{T}}P(0,\zeta )d\rho (\zeta )={\int }_{\mathbb{T}}1d\rho (\zeta )=\rho (\mathbb{T})$, because $P(0,\zeta )=1$ and the constant function $1$ is an admissible simple approximant in the definition of the integral.

Testing the boundary measure. For every $g\in C(\mathbb{T})$ with $g\ge 0$ one has ${\int }_{\mathbb{T}}gd\mu ={\lim }_{r\uparrow 1}{\int }_{\mathbb{T}}gd(u_{r}m)={\lim }_{r\uparrow 1}{\int }_{\mathbb{T}}g{u}_{r}dm\ge 0$: the first equality is the weak-star convergence recorded in step 2.1, the second uses the density-measure pairing $gd(u_{r}m)=g{u}_{r}dm$ of [L5], and for each $0\le r<1$ the function $g{u}_r$ is a bounded nonnegative Borel function on the probability space $(\mathbb{T},m)$, so step 1.1 gives ${\int }_{\mathbb{T}}g{u}_{r}dm\ge 0$; limits of nonnegative numbers are nonnegative.

The boundary measure is nonnegative. Define $\mathrm{\Lambda }(g):={\int }_{\mathbb{T}}gd\mu$ for real $g\in C(\mathbb{T})$; by step 3.1 these values are real, $\mathrm{\Lambda }$ is real-linear and bounded, and $\mathrm{\Lambda }(g)\ge 0$ whenever $g\ge 0$. By [L7], with Dependent Choice available from [L11], there is a finite regular Borel measure $\rho$ on $\mathbb{T}$ with $\mathrm{\Lambda }(g)={\int }_{\mathbb{T}}gd\rho$ for every real $g\in C(\mathbb{T})$. The complex-linear functionals $g\mapsto {\int }_{\mathbb{T}}gd\mu$ and $g\mapsto {\int }_{\mathbb{T}}gd\rho$ agree on real-valued functions and hence, by complex linearity, on all of $C(\mathbb{T},\mathbb{C})$; the uniqueness clause of [L6] therefore gives $\mu =\rho$. Thus $\mu$ is a nonnegative measure, and since $\mu$ is nonnegative and $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})=u(0)$ by step 2.1, also $\mu (\mathbb{T})=u(0)$.

Part (a). This proves (a): $u=P[\mu ]$ for the finite nonnegative regular Borel measure $\mu$ of step 4.1 with $\mu (\mathbb{T})=u(0)$, and if $\rho$ is any further finite nonnegative regular Borel measure with $u=P[\rho ]$, then $\rho$ is in particular a finite regular complex Borel measure representing $u$, so $\rho =\mu$ by the uniqueness clause of [L1] recorded in step 2.1. The converse direction (b) is step 2.2.

Normalized measures. For each $n$ the function $u_n$ is nonnegative harmonic with $u_n(0)=1$, so part (a) as proved in step 5.1 gives a unique finite nonnegative regular Borel measure ${\mu }_n$ on $\mathbb{T}$ with $u_n=P[{\mu }_n]$ and ${\mu }_n(\mathbb{T})=1$; in particular $\mathrm{\mid }{\mu }_n\mathrm{\mid }(\mathbb{T})=1$ for every $n$, so $({\mu }_n)$ is a sequence of probability measures.

Weak-star subsequence. By step 1.4 the space $C(\mathbb{T},\mathbb{C})$ is separable and the probability measures ${\mu }_n$ lie in the closed unit ball of its dual, so [L14] -- the ultrafilter lemma being available from the Axiom of Choice by [L11] -- provides a subsequence $({\mu }_{n_k})$ and an element of the dual which [L6] identifies with a finite regular complex Borel measure $\mu$ on $\mathbb{T}$ such that ${\int }_{\mathbb{T}}gd{\mu }_{n_k}\to {\int }_{\mathbb{T}}gd\mu$ for every $g\in C(\mathbb{T},\mathbb{C})$.

The limit measure is a probability measure. For real $g\in C(\mathbb{T})$ with $g\ge 0$ step 7.1 gives ${\int }_{\mathbb{T}}gd\mu ={\lim }_k{\int }_{\mathbb{T}}gd{\mu }_{n_k}\ge 0$, the inequality by step 1.1 applied to the finite measures ${\mu }_{n_k}$ and the bounded nonnegative Borel function $g$. The identification argument of step 4.1, with this positivity in place of step 3.1, now makes $\mu$ a nonnegative measure, and testing the constant function $1$ gives $\mu (\mathbb{T})={\lim }_k{\mu }_{n_k}(\mathbb{T})=1$, hence $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})=1$.

The limit function. Put $u:=P[\mu ]$. By the converse clause of [L1] the function $u$ is harmonic and lies in $h^1(\mathbb{D})$; by step 1.1 applied to the finite measure $\mu$ and the bounded nonnegative Borel function $P(z,\cdot )$ one has $u(z)={\int }_{\mathbb{T}}P(z,\zeta )d\mu (\zeta )\ge 0$ for every $z\in \mathbb{D}$; and $u(0)={\int }_{\mathbb{T}}P(0,\zeta )d\mu (\zeta )={\int }_{\mathbb{T}}1d\mu =\mu (\mathbb{T})=1$, exactly as in step 2.2.

Pointwise convergence. For each $z\in \mathbb{D}$ the function $\zeta \mapsto P(z,\zeta )$ is continuous on $\mathbb{T}$ by [L5], so the weak-star convergence of step 7.1 gives $u_{n_k}(z)={\int }_{\mathbb{T}}P(z,\zeta )d{\mu }_{n_k}(\zeta )\to {\int }_{\mathbb{T}}P(z,\zeta )d\mu (\zeta )=u(z)$.

Local uniform convergence. Fix a compact $K\subseteq \mathbb{D}$ and $\epsilon >0$. Uniform convergence on $K=\mathrm{\varnothing }$ is vacuous; assume $K\mathrm{\ne }\mathrm{\varnothing }$. By step 1.3 choose $\delta >0$ with $\mathrm{\mid }P(z,\zeta )-P(z^{\mathrm{\prime }},\zeta )\mathrm{\mid }<\epsilon$ whenever $z,z^{\mathrm{\prime }}\in K$ satisfy $\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }<\delta$ and $\zeta \in \mathbb{T}$, and by [L15] applied to the ambient balls $B(z,\delta )$ indexed by $z\in K$ choose $z_1,\dots ,z_m\in K$ with $K\subseteq {\bigcup }_{j=1}^{m}B(z_j,\delta )$. For $z\in K$ pick $j$ with $\mathrm{\mid }z-z_j\mathrm{\mid }<\delta$ and split $u_{n_k}(z)-u(z)={\int }_{\mathbb{T}}(P(z,\zeta )-P(z_j,\zeta ))d{\mu }_{n_k}(\zeta )+{\int }_{\mathbb{T}}P(z_j,\zeta )d({\mu }_{n_k}-\mu )(\zeta )+{\int }_{\mathbb{T}}(P(z_j,\zeta )-P(z,\zeta ))d\mu (\zeta )\mathrm{.}$ By [L10] together with $\mathrm{\mid }{\mu }_{n_k}\mathrm{\mid }(\mathbb{T})=1$ from step 6.1 and $\mathrm{\mid }\mu \mathrm{\mid }(\mathbb{T})=1$ from step 8.1, the first and third terms have modulus at most $\epsilon$, while the middle term tends to $0$ as $k\to \mathrm{\infty }$ by step 7.1 applied to the continuous function $P(z_j,\cdot )$. Hence ${\mathrm{lim\hspace{0.17em}sup}}_k{\sup }_{z\in K}\mathrm{\mid }{u}_{n_k}(z)-u(z)\mathrm{\mid }\le 2\epsilon +{\max }_{j\le m}{\lim }_k\mid {\int }_{\mathbb{T}}P(z_j,\zeta )d({\mu }_{n_k}-\mu )(\zeta )\mid \le 2\epsilon$, and since $\epsilon >0$ is arbitrary the subsequence converges to $u$ uniformly on $K$; as every compact subset of $\mathbb{D}$ arises this way, the convergence is locally uniform on $\mathbb{D}$.

Assembly. Steps 5.1, 2.2 and 6.1 through 11.1 prove the three assertions: (a) a nonnegative harmonic $u$ is $P[\mu ]$ for a unique finite nonnegative regular Borel measure $\mu$ with $\mu (\mathbb{T})=u(0)$; (b) conversely every finite nonnegative regular Borel measure gives a nonnegative harmonic $P[\mu ]$ with $P[\mu ](0)=\mu (\mathbb{T})$; (c) every sequence of nonnegative harmonic functions normalized by $u_n(0)=1$ has a subsequence converging locally uniformly on $\mathbb{D}$ to a nonnegative harmonic function with $u(0)=1$. The Axiom of Choice is used exactly as recorded: it supplies Dependent Choice and countable choice by [L11], for the Riesz representation [L6], the positive-functional lemma [L7] and the countable dense family [L13], and it supplies the ultrafilter lemma for [L14]; no other choice was made. ∎