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Suppose $E\subset \mathbb{R}^d$ is a measurable set with finite measure $m(E) < \infty$. Let $f_n: E \to \mathbb{R}$ be a sequence of nonnegative measurable functions such that $f_n(x) \to f(x)$ almost everywhere in $E$. Prove that $$\lim\limits_{n \to \infty}\int_E \frac{f_n(x)}{1+f_n(x)}dx = \int_E \frac{f(x)}{1+f(x)}dx.$$ I'm trying to prove this equality in preparation for an exam of real analysis. I would appreciate your help in solving this problem. Thank you.

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Hint: $$ \frac{f_n(x)}{1+f_n(x)} \leq 1, $$ and since $m(E)<\infty$ we can pass to the limit (why?)

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  • $\begingroup$ I believe we can pass to the limit by the bounded convergence theorem. Thanks for the hint. $\endgroup$ – sergio Sep 19 '16 at 17:16

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