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Math Help - Lebesgue Integral

  1. #1
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    Lebesgue Integral

    Let (f_n) be an increasing sequence of integrable function such that (f_n) \rightarrow f on X and f is integrable.
    Show that \int_X f = \lim_n \int_X f_n.

    Can anyone help?
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  2. #2
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    Since f_n(x)\longrightarrow f(x) and \vert f_n(x) \vert \leq \vert f(x) \vert and f is integrable, then by the Dominated convergence theorem the result follows. This is valid if X \subset \mathbb{R} ^n with Lebesgue measure. I don't know if the DCT holds for arbitrary measure spaces.
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