Beppo Levi's theorem

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Let $f_n$ is a sequence of integrable functions and $\sup_n\int f_n d\mu<\infty$. I need to show that, if $f_n\uparrow f$, then $f$ is integrable and $\int f_nd\mu \rightarrow \int fd\mu$. This is claimed to be named as "Beppo Levi's theorem".

It is not stated that $f_n$ are non-negative or non-negative a.e., so it is clearly not the same assumptions of Monotone convergence theorem. I have no clue how to show the convergence of the integral in case of the above assumptions, and not the non-negativeness assumption.

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1 Answer

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Define $$g_n=f_{n}-f_1$$

Then $g_n$ is an increasing sequence of non-negative integrable functions, and $g_n\to g:=f-f_1$, thus by monotone convergence theorem $$\int g_n d\mu\to\int g d\mu$$

Thus the result follows.

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