# Beppo Levi's theorem, is this assertion correct?

My notes report the following assertion for the theorem:

Beppo Levi's Theorem: Let $$E$$ be a measurable set and $$\{ f_n(x)\}$$ a sequence of integrable functions on E, such that $$\lim\limits_{n\to\infty} f_n(x) = f(x)$$ (pointwise convergence) almost everywhere on E, and $$f_n(x)\leq f(x)$$. Then $$f(x)$$ is integrable on E and $$\lim\limits_{n\to\infty} \int\limits_E f_n(x) = \int\limits_E f(x)$$

Is this correct? Cause my book reports multiple versions of the theorem, but not this one.

• What does punctual convergence mean? Jan 16, 2019 at 15:17
• Pointwise, my bad, translation error. Let me correct it Jan 16, 2019 at 15:18
• It should be only $\lim\limits_{n\to\infty} f_n(x) = f(x)$ without "for every $n$" because $n$ is already in the limit sign. Jan 16, 2019 at 15:20
• I don't know why, but my notes report both. I'll take your tip anyway! Jan 16, 2019 at 15:21
• As is, the statement is wrong, because $E=\mathbb{R}^+$, $f_n=1_{[-n,n]}$ seems to give your theorem a problem. Jan 16, 2019 at 15:30

First this theorem is wrong

Consider $$E = \mathbb R$$, $$f=0$$ and $$f_n = -\chi_{[n,n+1]}$$. Where $$\chi_{[n,n+1]}$$ is the indicator function of the interval $$[n,n+1]$$.

They satisfy the hypothesis, but the conclusion doesn't hold as

$$-1 = \int_{\mathbb R} f_n \neq \int_{\mathbb R} f = 0$$ while $$(f_n)$$ converges pointwise to the always vanishing function $$f$$.

Second this would more related to dominated convergence theorem

See Dominated convergence theorem if the hypothesis would be correctly set.

The theorem is usually known as the monotone convergence theorem and comes with an extra requirement that $$f_1(x)\le f_2(x)\le\dots \le f(x)$$ for almost every $$x$$. Other than this the other parts of your statement is correct.

• My notes report that the classical statement requires that $f_1(x) < f_2(x) < \dots < f_n(x) < \dots$ but it can be proved not to be necessary... Isn't that right? Jan 16, 2019 at 15:26
• That’s not the monotone convergence theorem. There is no assumption on the sequence $f_n$. Jan 16, 2019 at 15:27
• @Bafforasta Then you'd need extra assumptions. As it is now the statement is not true. Jan 16, 2019 at 15:30
• @Mindlack I know the statement in the OP is not the MCT that's why I said that it missed one assumption, i.e. monotonicity. The reason being the name Beppo Levi is associated with the MCT so I made the link. Jan 16, 2019 at 15:35
• @BigbearZzz I just found this post: math.stackexchange.com/questions/1177788/beppo-levis-theorem this looks pretty similar Jan 16, 2019 at 15:46