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Can anyone think of a sequence of functions $f_n:[0,\infty) \to \mathbb{R}$ such that $f_n \to f$ uniformly but $\int_0^\infty f_n \nrightarrow \int_0^\infty f$ ?

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up vote 4 down vote accepted

Take the constant functions $f_n=\frac{1}{n}$. They converge uniformly to zero, but $\int_0^\infty f_n =\infty$, and $\int_0^\infty f=0$.

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One can even find an example using a sequence of improperly integrable functions, take $f_n={1\over n}\chi_{[0,n]}$, where $\chi_A$ is the indicator function on the set $A$.

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