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Monotone Convergence

  1. Apr 18, 2012 #1
    1. The problem statement, all variables and given/known data

    16jhg6u.jpg

    2. Relevant equations

    Monotone Convergence Theorem:

    http://img696.imageshack.us/img696/5469/mct.png [Broken]

    3. The attempt at a solution

    I know this almost follows from the theorem. But I first need to write [itex]\displaystyle \int_{I_n} f = \int_S f_n[/itex] for some [itex]f_n[/itex] in such a way that [itex](f_n)[/itex] is an increasing sequence tending to [itex]f[/itex]. (Then we have something that satisfies the hypotheses of the theorem.) What [itex]f_n[/itex] could I use?

    Then in the case of any function [itex]g[/itex] can I consider positive and negative parts?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Apr 19, 2012 #2
    Hmm, what if you let [itex]f_{n} \left( x \right) = \left\{ \begin{array}{rl} f \left( x \right) &, x \in I_{n} \\ 0 &, x \not \in I_{n} \end{array} \right.[/itex]. I'm not sure if [itex]f_{n} \in \mathcal{L}^{1} \left( \mathbb{R}^{k} \right)[/itex] but it is an increasing sequence of functions which converges point-wise to [itex]f[/itex].
     
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