Precisely, I am working on the monotone convergence theorem from the book of Folland, Real analysis. Statement is as follows:

If $\{f_n\}$ is a sequence in $L^+$ such that $f_j \leq f_{j+1}$ for all $j$ and $\displaystyle f = \lim_{n \to \infty} f_n$ $\displaystyle(=\sup_n f_n)$, then $\displaystyle\int f = \lim_{n \to \infty} \int f_n$.

Then, I did not understand what does it mean taking the limit of the sequence of functions. Also, I did not understand that how it equals to the supremum. Thanks.

  • 1
    $\begingroup$ This is immediately related to Dini's Theorem. $\endgroup$ – Pedro Tamaroff Nov 7 '12 at 19:55
  • $\begingroup$ Thanks Peter, it was useful. $\endgroup$ – user48547 Nov 7 '12 at 20:01
  • $\begingroup$ @PeterTamaroff: What is the "immediate" relation? I see that both involve monotone (pointwise) convergence, but is there any connection beyond that? $\endgroup$ – Jesse Madnick Dec 9 '12 at 1:40

The limit is here pointwise, i.e. $f$ is the function such that $f(x)=\lim_{n\to\infty} f_n(x)$ holds for all $x$. By assumption of monotony, the limit is also the supremum (again pointwise).

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    $\begingroup$ Smells like measure theory. It could be that $f$ is only the pointwise a.e. limit of the $f_n$. $\endgroup$ – kahen Nov 7 '12 at 19:44

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