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.


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