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Math Help - Are there any examples of...

  1. #1
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    Are there any examples of...

    unbounded functions that are Lebesgue integrable?
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  2. #2
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    Yes, for example f(x) = 1/\sqrt x on (0,1).

    [If you want a proof that f is Lebesgue integrable, let f_n(x) = \begin{cases}n&(0<x\leqslant1/n^2),\\1/\sqrt x&(1/n^2<x<1).\end{cases} Note that each f_n is integrable (with integral 2 (1/n)) and that (f_n) increases to f, and use the monotone convergence theorem.]
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