Can you solve the Problem of the Week #41 - March 11th, 2013?

[Chris L T521]

Here's this week's problem.

-----

Problem: Suppose that $\{f_n\}$ is a collection of non-negative measurable functions with $f_1\geq f_2\geq\cdots\geq 0$ and $f_n(x)\rightarrow f(x)$ for every $x\in X$. Furthermore, suppose that $f_1\in L_{\mu}^1(X)$. Prove that $f\in L_{\mu}^1(X)$ and
\[\int_X f\,d\mu = \lim_{n\to\infty} \int_X f_n\,d\mu.\]

-----

 

[Chris L T521]

This week's question was correctly answered by girdav. You can find his solution below.

Spoiler

As $0\leqslant f(x)\leqslant f_1(x)$ for all $x$, $f$ is integrable. Define $g_n:=f_1-f_n$: this forms a non decreasing sequence of measurable functions. Hence we can apply monotone convergence theorem, which will yield the result.

It's actually a reversed version of the MCT.
Note that we can relax the assumption "for all $x$" considering only "for almost every $x$" with the underlying measure.