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Spring 2019 - Problem 2

measure theory, weak convergence

Let $\mu$ be a Borel probability measure on $\br{0, 1}$ that has no atoms (this means that $\mu\p{\set{t}} = 0$ for any $t \in \br{0, 1}$ ). Let also $\mu_1, \mu_2, \ldots$ be Borel probability measures on $\br{0, 1}$ . Assume that $\mu_n$ converges to $\mu$ in the weak-* topology. Denote $F\p{t} = \mu\p{\br{0, t}}$ and $F_n\p{t} = \mu_n\p{\br{0, t}}$ for each $n \geq 1$ and $t \in \br{0, 1}$ . Prove that $F_n$ converges uniformly to $F$ .

Solution.

Let $\set{f_n}_n, \set{g_n}_n$ be continuous functions such that $f_n \leq \chi_{\br{0,t}} \leq g_n$ for all $n$ and so that $f_n$ increases to $\chi_{\pco{0,t}}$ and $g_n$ decreases to $\chi_{\br{0,t}}$ by taking piecewise linear functions. Then

\begin{gathered} \int f_k \,\diff\mu_n \leq \mu_n\p{\br{0, t}} \leq \int f_k \,\diff\mu \\ \implies \int f_k \,\diff\mu \leq \liminf_{n\to\infty} \mu_n\p{\br{0,t}} \leq \limsup_{n\to\infty} \mu_n\p{\br{0,t}} \leq \int g_k \,\diff\mu, \end{gathered}

by weak-* convergence. We can then apply monotone convergence on the left-most integral and dominated convergence on the right-most integral to get

\mu\p{\pco{0, t}} \leq \liminf_{n\to\infty} \mu_n\p{\br{0,t}} \leq \limsup_{n\to\infty} \mu_n\p{\br{0,t}} \leq \mu\p{\br{0, t}}.

Since $\mu$ has no atoms, these are all equalities, so $F_n$ converges pointwise to $F$ . To prove pointwise convergence, let $\epsilon > 0$ . Since $F$ is continuous, it is uniformly continuous, so let $\delta > 0$ be as in the definition of uniform continuity. Then by compactness, there exist $0 = t_0 < t_1 < \cdots < t_N = 1$ such that the $B\p{t_i, \delta}$ cover $\br{0, 1}$ . By pointwise convergence, there exists $M \in \N$ such that for $n > M$ , $\abs{F_n\p{t_i} - F\p{t_i}} < \epsilon$ for all $1 \leq i \leq N$ . Thus, given $t \in \br{0, 1}$ , there exists $i$ such that $t \in \br{t_i, t_{i+1}}$ , and so by monotonicity of every function,

\begin{aligned} \abs{F_n\p{t} - F\p{t}} &\leq \abs{F_n\p{t} - F_n\p{t_i}} + \abs{F_n\p{t_i} - F\p{t_i}} + \abs{F\p{t_i} - F\p{t}} \\ &\leq F_n\p{t} - F_n\p{t_i} + 2\epsilon \\ &\leq F_n\p{t_{i+1}} - F_n\p{t_i} + \p{F\p{t_{i+1}} - F\p{t_i}} + 2\epsilon \\ &\leq \abs{F_n\p{t_{i+1}} - F\p{t_{i+1}}} + \abs{F_n\p{t_i} - F\p{t_i}} + 2\epsilon \\ &\leq 4\epsilon, \end{aligned}

so the convergence is uniform.