If you have a sequence of integrable functions [itex]\{f_n(x)\}[/itex] on [itex][0,1][/itex] which converges to a function [itex]f(x)[/itex] pointwise for every [itex]x\in [0,1][/itex] that has the following properties:(adsbygoogle = window.adsbygoogle || []).push({});

(1) [itex]0 \leq f_n(x) \leq f(x)[/itex] for every n and every x; and

(2) [itex]\int_0^1 f_n(x)dx = 1[/itex] for every n;

does it necessarily follow that the limit function [itex]f[/itex] is integrable and satisfies [itex]\int_0^1 f(x) dx = 1[/itex]?

I can't think of why this would need to be true using the standard Lebesgue convergence theorems (bounded, monotone, or dominated), since none of them seem to apply. But I can't think of a counterexample to save the life of me. Can anyone help?

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# Sequence of integrable functions on [0,1]

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