# Measure Theory (A counterexample to interchanging limits and integration)

(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.

Rosetta

59

## Answer

**Answers can only be viewed under the following conditions:**

- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.

1 Attachment

Ering

93

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.

- answered
- 198 views
- $40.00

### Related Questions

- Convergence integrals
- Convergence and Integrability of Function Series in Measure Spaces and Applications to Series Expansion Integrals
- What is the Lebesgue density of $A$ and $B$ which answers a previous question?
- Sigma-Algebra Generated by Unitary Subsets and Its Measurable Functions
- Question on a pre-measure defined by Folland's real analysis book
- Summation of Catalan Convolution
- How do we describe an intuitive arithmetic mean that gives the following? (I can't type more than 200 letters)
- Subsets and Sigma Algebras: Proving the Equality of Generated Sigma Algebras