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Let $f:X \to \mathbb R$ be a function s.t. : There are integrable $(f_n)_n$ and $(g_n)_n$ s.t. $g_n \leq f \leq f_n$ and $\int f_n \to c; \int g_n \to c$ for some real $c$. Then I want to show that $f$ is integrable. I would appreciate some hints.

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1 Answer 1

Let $$ F(x)=\inf_{n\in\mathbb N}f_n(x), \quad G(x)=\sup_{n\in\mathbb N}g_n(x). $$ Clearly, $F,G$ are measurable functions and $$ G(x)\le f(x)\le F(x). $$ Also, due to Fatou's Lemma (the 2nd inequality) $$ 0\le \int F-G =\int \inf_{n\in N}f_n-g_n \le \inf_{n\in\mathbb N}\int f_n-g_n=\lim_{n\to \mathbb N}\int f_n-g_n=0. $$ Thus if we set $d(x)=F(x)-G(x)\ge 0$, then $\int d=0$, and hence $d(x)=0$, a.e. Thus $f(x)=F(x)=G(x)$, a.e., and therefore $f$ is measurable.

Now $f_n-g_n\ge f_n-f\ge 0$, and as $f_n-g_n$ is integrable, so is $f_n-f$, and so is $f$, since $f_n$ is integrable. Finally, $$ \int g_n\le \int f\le \int f_n, $$ implies, letting $n\to\infty$, that $\int f=c$.

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