If $ f:\mathbb{R^n} \rightarrow K $, where $K = \mathbb{R}$ or $\mathbb{C}$, is such that $f= \lim_{v \rightarrow \infty} f_v$ a.e., where $\lbrace f_v\rbrace $ is a Cauchy sequence in $L_1$ of continuous compact support functions, then $f $ is integrable.
I am trying to use the fact that the set $C_{c}^{\infty}$ is dense in $L_p$, but I don't see how to connect this with the given hypothesis. Any hint please?
$L^{1}$ is complete. Hence there exists $g \in L^{1}$ such that $\int |f_n-g| \to 0$. But then three is a subsequence $(f_{n_k})$ convergeing a.e. to $g$. It follows that $f=g$ a.e., so $f \in L^{1}$.