# Arzelà's Dominated Convergence Theorem for improper integrals?

Consider the Arzela's Dominated Convergence Theorem for Riemann integrals:

Take a function $f_n$ defined on a bounded and closed interval $[a,b]$ and Riemann integrable on $[a,b]$ $\forall n \in \mathbb{N}$. Let $|f_n(x)|\leq M$ with $M>0$ $\forall x \in [a,b]$. Let $\lim_{n\rightarrow \infty}f_n(x)=f(x)$ $\forall x \in [a,b]$ with $f$ Riemann integrable on $[a,b]$. Then, $$\lim_{n\rightarrow \infty}\int_a^b |f_n(x)-f(x)|dx=0$$

Question: Is there a version of this theorem for improper integrals? Specifically, I am looking for something like