Extras

Compute the following limits:

• $$\lim_{n\to\infty} \sum_{k\geq 1} {1\over k^2} \sin^n(k)$$
• $$\lim_{n\to\infty} \sum_{k\geq 1} {1\over k} e^{-k/n}$$

For the first, use that \begin{align*} {\left\lvert { \sum_{k\geq 1} {1\over k^2} \sin^n(k) } \right\rvert} \leq \sum_{k\geq 1} {\left\lvert { {1\over k^2} \sin^n(k) } \right\rvert} \sum_{k\geq 1} {\left\lvert { {1\over k^2}} \right\rvert} < \infty ,\end{align*} since $${\left\lvert {\sin(x)} \right\rvert} \leq 1$$ and $$x^n < x$$ for $${\left\lvert {x} \right\rvert}\leq 1$$. By the dominated convergence theorem, we can pass the limit inside. Using the same fact as above, $$\lim_{n\to\infty}\sin^n(x) = 0$$,

For the second, the claim is that it diverges (very slowly). Note that $$\lim_{n\to\infty} e^{-k/n} = 1$$ for any $$k$$. By Fatou, we have \begin{align*} \liminf_{n\to\infty} \sum_{k\geq 1} {e^{-k/n} \over k} \geq \sum_{k\geq 1} \liminf_{n\to\infty} {e^{-k/n} \over k} = \sum_{k\geq 1} {1 \over k} = \infty .\end{align*}

Let $$(\Omega,{\mathcal{B}})$$ be a measurable space with a Borel $$\sigma{\hbox{-}}$$algebra and $$\mu_n: {\mathcal{B}}\to [0, \infty]$$ be a $$\sigma{\hbox{-}}$$additive measure for each $$n$$. Show that the following map is again a $$\sigma{\hbox{-}}$$additive measure on $${\mathcal{B}}$$: \begin{align*} \mu(B) \coloneqq\sum_{n\geq 1} \mu_n(B) .\end{align*}

Apply Fubini-Tonelli to commute two sums: \begin{align*} \mu\qty{\bigcup_{1\leq k \leq M} E_k}\coloneqq &= \sum_{n\geq 1} \mu_n\qty{\bigcup_{1\leq k \leq M} E_k}\\ &= \sum_{n\geq 1} \sum_{1\leq k \leq M} \mu_n\qty{E_k}\\ &= \sum_{1\leq k \leq M}\sum_{n\geq 1} \mu_n\qty{E_k} \text{FT} \\ &\coloneqq\sum_{1\leq k \leq M} \mu(E_k) .\end{align*}