Find a necessary and sufficient condition for the integrabiity of a function

Question

Let $\{ a_n \}$ be a sequence of real numbers. let $f= \sum \limits_{n=1}^\infty a_n\chi_{[n,n+1)}$.

if $\sum \limits_{n=1}^\infty a_n$ is convergent, is $f$ lebesgue integrable?

find a necessary and sufficient condition for integrability of $f$

I spent a good amount of time thinking about this problem. but I cannot solve this. can anyone give me a help?

Answer 1

Hints:

1. Since the intervals $[n,n+1)$ are disjoint, it holds that $$|f| = \sum_{n =1}^{\infty} |a_n| \chi_{[n,n+1)}.$$
2. Conclude that $$\int |f| = \sum_{n=1}^{\infty} |a_n|$$ (e.g. using monotone convergence). This gives you a sufficient and necessary condition for integrability of $f$.
3. To show that convergence of $\sum_{n=1}^{\infty} a_n$ is not sufficient, pick your favourite sequence $(a_n)_{n \in \mathbb{N}}$ such that $\sum_{n=1}^{\infty} a_n$ converges but $\sum_{n=1}^{\infty} |a_n|=\infty$ (e.g. $a_n = (-1)^n/n$).