# Need help proving that $\int_{\bigcup_{n=1}^{\infty}E_n}f{\rm d}\lambda = \lim_{n\to\infty}\int_{E_n}f{\rm d}\lambda$

Here is the problem: If $f$ is a nonnegative Lebesgue measurable function and $\{E_n\}_{n=1}^{\infty}$ is a sequence of Lebesgue measurable sets with $E_1\subset E_2\subset ...$, then

$\int_{\bigcup_{n=1}^{\infty}E_n}f{\rm d}\lambda = \lim_{n\to\infty}\int_{E_n}f{\rm d}\lambda$.

I know that $\int_{\bigcup \limits_{n=1}^{\infty} E_{n}} f \,d\lambda = \int \limits_{\Bbb R} f \chi_{\bigcup \limits_{n=1}^{\infty} E_{n}} \,d\lambda$.

I'm also thinking that I should apply the Monotone Convergence Theorem.

However, I don't know where to go from here. Any help would be appreciated.

• – Namaste Feb 25 '18 at 0:09
• DO NOT ask the same question TWICE. – Namaste Feb 25 '18 at 0:11
• How could you not know it was already asked, when you asked the question I link, and ask it again here?: NOT okay. – Namaste Feb 25 '18 at 0:45
• "Here is the following problem: Suppose that f is Lebesgue integrable over E and that $\{E_{n}\}_{n=1}^{\infty}$ is a sequence of lebesgue measurable sets with $E_1$ $\subset$ $E_2$ $\subset$.... and $\cup_{n=1}^{\infty} (E_{n})$ = E. Prove that $\int_E f\,d\lambda$ = $\lim_{n \to \infty} \int_{E_n} f\,d\lambda$. Any suggestions/ hints on how to start this problem would be appreciated." Asked already by you here – Namaste Feb 25 '18 at 0:48
• Hold up....these are two different questions. I simply thought you were referring to a different question (that was similar to this question) asked by someone else. – FoxViking Feb 25 '18 at 0:49

Write $\displaystyle\int_{\bigcup_{n=1}^{\infty}E_{n}}fd\lambda=\int_{X}\chi_{\bigcup_{n=1}^{\infty}E_{n}}fd\lambda$ and $\displaystyle\int_{E_{n}}fd\lambda=\int_{X}\chi_{E_{n}}fd\lambda$ and now observe that $f\chi_{E_{n}}\uparrow f\chi_{\bigcup_{n=1}^{\infty}E_{n}}$ so Monotone Convergence Theorem applies here.
• $f\chi_{E_{n}}\uparrow f\chi_{\bigcup_{n=1}^{\infty}E_{n}}$ follows since f is a nonnegative Lebesgue measurable function and $\{E_n\}_{n=1}^{\infty}$ is a monotone nondecreasing sequence of Lebesgue measurable sets? – FoxViking Feb 24 '18 at 23:52