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Given a sequence of nonnegative functions $(f_n)_n$ converging almost surely (for the Lebesgue measure $d\mu$) to $f$. Assume that $\int f_n d\mu \rightarrow c < \infty$ as $n$ goes to infinity. Prove $\int f d\mu$ is defined and in $[0,c]$, but does not necessarily have to be $c$.

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up vote 2 down vote accepted

For a counterexample, let $f_n$ be defined on $[0,1]$, and its graph is a line segment from $(0,n)$ to $(1/n,0)$ and $f_n(x)=0$ for $x\ge 1/n$.

Then $\int f_n=1/2$, and $f_n\to 0$ except in $0$.

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