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Thread: Monotone Convergence

  1. #1
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    Monotone Convergence



    Monotone Convergence Theorem:


    I know this almost follows from the theorem. But I first need to write $\displaystyle \displaystyle \int_{I_n} f = \int_S f_n$ for some $\displaystyle f_n$ in such a way that $\displaystyle (f_n)$ is an increasing sequence tending to $\displaystyle f$. (Then we have something that satisfies the hypotheses of the theorem.) What $\displaystyle f_n$ could I use?
    Then in the case of any function $\displaystyle g$ can I consider positive and negative parts?
    Last edited by ProofbyInduction; Apr 18th 2012 at 02:10 PM.
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  2. #2
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    Re: Monotone Convergence

    Define $\displaystyle f_n(x)= f(x)$ if $\displaystyle x\in I_n$, $\displaystyle f_n(x)= 0$ otherwise.
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