Fall 2020 - Problem 3

Solution.

1. By Stone-Weierstrass, the set

   $$A = \set{\sum_{k=1}^n f_k\p{x}g_k\p{y} \st n \in \N,\ f_k, g_k \in C\p{\br{0,1}}}$$

   is dense in $C\p{\br{0,1}^2}$. Hence, by applying weak convergence and Fubini's theorem, we see that $\mu_n \otimes \mu_n$ converges weakly-* to $\mu \otimes \mu$.

2. Let $\mu_n = \delta_0$ if $n$ is odd and $\mu_n = \frac{1}{n} \sum_{k=1}^n \delta_{k/n^2}$ for $n$ even. Then for $n$ even and $g$ is continuous,

   $$\int g\p{x} \,\diff\mu_n\p{x} = \frac{1}{n} \sum_{k=1}^n g\p{\frac{k}{n^2}} \xrightarrow{n\to\infty} g\p{0},$$

   by continuity, and it's clear that the same holds for the odd terms. However, for $n$ od, we have

   $$\int_{\set{0 \leq x \leq y \leq 1}} \diff\mu_n\p{x} \,\diff\mu_n\p{y} = 0$$

   and for $n$ even,

   $$\begin{aligned} \int_{\set{0 \leq x \leq y \leq 1}} \diff\mu_n\p{x} \,\diff\mu_n\p{y} &= \int_0^1 \int_0^y \diff\mu_n\p{x} \,\diff\mu_n\p{y} \\ &= \sum_{j=1}^{n^2} \int_{\pco{\frac{j-1}{n^2}, \frac{j}{n^2}}} \int_0^y \diff\mu_n\p{x} \,\diff\mu_n\p{y} \\ &= \sum_{j=1}^n \int_{\pco{\frac{j-1}{n^2}, \frac{j}{n^2}}} \frac{j}{n} \,\diff\mu_n\p{y} + \sum_{j=n+1}^{n^2} \int_{\pco{\frac{j-1}{n^2}, \frac{j}{n^2}}} \frac{n}{n} \,\diff\mu_n\p{y} \\ &= \sum_{j=1}^n \frac{j}{n^2} \\ &= \frac{n + 1}{2n} \xrightarrow{n\to\infty} \frac{1}{2}, \end{aligned}$$

   so the sequence diverges.