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Prove monotone convergence theorem when $int |f_1| dmu < infty$ holds

Mathematics Asked by Evan Kim on January 26, 2021

This question is from 3.A exercise 20 from Axler’s MIRA book.

Suppose $(X,S,mu)$ is a measure space and $f_1, f_2, dots$ is a monotone sequence of $S$-measurable functions. Define $f: X to [-infty,infty]$ by $$f(x) = lim_{ktoinfty}{f_k(x)}.$$ Prove that if $int|f_1|dmu < infty$, then $$lim_{ktoinfty} int{f_k(x)}dmu = int fdmu. $$

The part that I don’t understand how to use the assumption $int |f_1| < infty$. For example if we assumed that $int |f_1| = infty$, then I guess a counter example could be found to show why the equality won’t hold? I don’t think I am thinking correctly about this problem at all and my intuition is all off.

One Answer

The assumption $int |f_1|dmu<infty$ is necessary for some cases, e.g., when your sequence is decreasing and non-negative. For example, consider $f_n=chi_{[n,infty)}$ over $mathbb{R}$ with the usual Lebesgue measure. Then $int_mathbb{R} f_ndmu=infty$, but $lim f_n=0$, so the statement does not hold.

You may follow the hint below to solve your problem:

Answered by Hanul Jeon on January 26, 2021

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