Measure Theory (A counterexample to interchanging limits and integration)

Consider the probability space $(\mathbb{R}, \mathcal{B}, \mathrm{P})$ with $\mathrm{P}$ being the Cauchy distribution.

(a) Let $f_n(x)=|x| \mathbb{1}_{[-n, n]}$. Show that $\lim _{n \rightarrow \infty} \int_{\mathbb{R}} f_n d \mathrm{P}=\int_{\mathbb{R}}\left(\lim _{n \rightarrow \infty} f_n\right) d \mathrm{P}$. What is the value of this expression?
(b) Let $f_n(x)=x \mathbb{1}_{[-n, n]}$. Show that $\lim _{n \rightarrow \infty} \int_{\mathbb{R}} f_n d \mathrm{P} \neq \int_{\mathbb{R}}\left(\lim _{n \rightarrow \infty} f_n\right) d \mathrm{P}$. Argue that this does not contradict the Monotone Convergence Theorem.

Answer

Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer

1 Attachment

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.