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Let $\{f_n\}$ be a sequence of nonnegative measurable functions on $E$ that converges pointwise on $E$ to $f$. Suppose $f_n \leq f$ on $E$ for each $n$. Show that $\lim\limits_{n \to \infty} \int_E f_n=\int_E f$

This can almost use the Monotone Convergence Theorem, but the sequence is nonnegative. Can I just choose a subsequence, $\{f_{n_k}\}$ of $\{f_n\}$ that is increasing or is there a way to construct an increasing sequence.?

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  • $\begingroup$ There is a way to construct an increasing sequence. As a hint, I suggest using a simple construction to define a sequence $g_m$ of nonnegative functions with the following properties: (1) $g_m$ is an increasing sequence, (2) $g_m$ converges to $f$ pointwise, and (3) $g_m$ depends only on the functions $f_n$ for $n\geq m$. If you've constructed the $g_m$ in the right way, what you want to show follows from the monotone convergence theorem applied to the $g_m$. $\endgroup$
    – froggie
    Nov 12, 2012 at 2:49
  • $\begingroup$ But What gurantees that $lim∫f^{(n)}=lim∫g^{(n)}$ $\endgroup$
    – Amr
    Nov 12, 2012 at 2:51
  • $\begingroup$ @Amr: This will follow from the construction of the $g_m$. Maybe I should make that more clear: I'm not claiming that any $g_m$ satisfying (1),(2), and (3) will solve the problem. I have a specific (and simple) construction in mind. $\endgroup$
    – froggie
    Nov 12, 2012 at 2:55
  • $\begingroup$ Emm. OK. I think you meant $g^{(n)}$ =inf{$f^{(n)},f^{(n+1)},...$} $\endgroup$
    – Amr
    Nov 12, 2012 at 2:55

1 Answer 1

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Let $g_n = \inf_{m \geq n} f_m$. Then the $g_n$ are increasing to $f$ pointwise and $g_n \leq f_n \leq f$. Apply MCT.

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