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Let $f:\mathbb{R}\rightarrow\bar{\mathbb{R}}$ Lebesgue integrable. Prove that for $\epsilon >0$ there exists a finite interval $[a,b]$ such that

$$\left|\int{f(x)}dx-\int_{a}^b f(x)dx\right|<\epsilon.$$

My attempt: If $f$ is integrable on $[a,b]$, then for any $\epsilon > 0$ there exists $\delta > 0$ such that for any measurable set $D \subset [a,b]$ with measure $\mu(D) < \delta$ we have

$$\left|\int_{a}^b f(x)dx\right|<\epsilon/2.$$

This is where I'm stuck. Can someone help me?

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    $\begingroup$ Apply Lebesgue Dominated Convergence Theorem to $\{\chi_{[-n,n]} f\}_{n \in \mathbb{N}}$. $\endgroup$ Jul 18, 2019 at 22:56
  • $\begingroup$ I don’t understand.... The first integral is indefinite, so the difference will be a constant of integration plus some other value $\endgroup$ Jul 19, 2019 at 0:23

2 Answers 2

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You can easily show via dominated convergence that $$ \lim_{n\to\infty} \int_{-n}^n f(x)dx=\int f(x)dx. $$ By definition, for a fixed $\epsilon>0$, this means that for all $N$ large enough, $$ \bigg\vert \int f(x)dx-\int_{-N}^N f(x)dx\bigg\vert<\epsilon. $$

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If $$A=\bigcup_{n=1}^{\infty}A_n$$ and $$A_1\subset A_2\subset\dots,$$ then $$\int_A f\,\text{d}\mu=\lim\limits_{n\to\infty}\int_{A_n} f\,\text{d}\mu.$$ This is the fundamental property of the Lebesgue integral.

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  • $\begingroup$ I wouldn't call this the fundamental property of the Lebesgue integral. $\endgroup$ Jul 18, 2019 at 23:02
  • $\begingroup$ What's your problem? $\endgroup$
    – szw1710
    Jul 18, 2019 at 23:04
  • $\begingroup$ Well, this is obviously a matter of opinion, but this wording (with the definite article) suggests that this property somehow captures the essence and meaning of the Lebesgue integral which I wouldn't quite agree with. $\endgroup$ Jul 18, 2019 at 23:07
  • $\begingroup$ So, this is the ''language problem''. If you are thinking in this way, I agree, this is not the essence of the Lebesgue integral. However, to make this discussion clear for the reader, I leave the form I used. $\endgroup$
    – szw1710
    Jul 18, 2019 at 23:10

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