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Thread: Lebesgue Integral

  1. #1
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    Lebesgue Integral

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

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