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Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, what is a counterexample?

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$f_n=1_{[n,\infty)}$ is a counterexample.

Another one is $f_n=1_{[n,n+1)}$.

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