Let $\mu$ be the Lebesgue measure on $\mathbb{R}$ and denote by $\chi_A$ the characteristic function of $A\subseteq\mathbb{R}$. Assume that $a_n\in\mathbb{R}$ and $I_n\subseteq\mathbb{R}$ is a bounded nonempty half-open interval of type $[a,b)$ for each $n\in\mathbb{N}$ such that $\sum_{n=0}^\infty|a_n|\mu(I_n)$ converges.
Is it true that the function $\sum_{n=0}^\infty a_n\chi_{I_n}$ converges pointwise almost everywhere? If so, how to prove it?
Two observations:
if $\mu(I_n)$ were bounded below by a positive constant, say $m$, then $\sum_{n=0}^\infty|a_n|\mu(I_n)\geq m\sum_{n=0}^\infty|a_n|$ so that $\sum_{n=0}^\infty|a_n|$, $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty a_n\chi_{I_n}$ would converge (the last everywhere)
$\sum_{n=0}^\infty a_n\chi_{I_n}$ may fail to converge at a point. Example: $a_n=1$ and $I_n=[0,(n+1)^{-2})$ so that $\sum_{n=0}^\infty |a_n|\mu(I_n)<\infty$ but $\sum_{n=0}^\infty a_n\chi_{I_n}(0)=\sum_{n=0}^\infty 1$ diverges.
By Tonelli's Theorem $\int \sum |a_n| \chi_{I_n} d\mu = \sum |a_n| \int \chi_{I_n} d\mu=\ \sum |a_n|\mu(I_n) <\infty$. This implies that $\sum |a_n|\chi_{I_n} <\infty$ almost everywhere so the series converges absolutely at almost all points.
PS Tonelli's Theorem can be replaced by Monotone Convergence Theorem or Fatou's Lemma.