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My friend and I came upon a problem in Real Analysis. It called for a sequence of Lebesgue integrable functions $(f_n)$ converging everywhere to a Lebesgue integrable function $f$ such that

$$ \lim_{n \to \infty} \int_{-\infty}^{+\infty} \! f_n(x) \, \mathrm{d}x < \int_{-\infty}^{+\infty} \! f(x) \ $$

Unfortunately, we haven't had much luck finding any examples. Does anyone know of any?

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  • $\begingroup$ $f_n=-n\chi_{[0,1/n]}$. $\endgroup$ Oct 31 '14 at 10:47
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let $f_n(0)=-n$, $f(x)=0, \forall x\in(-\infty,0)\cup[\frac{1}{n},\infty)$, linear between $(0,\frac{1}{n})$, it converges everywhere to $f=-\infty\chi_{\{0\}}$, which has $0$ integral.

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