In Serge Lang's Real and Functional Analysis, first part of Lemma 3.1 (p.129) states
Let $\{f_n\}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise almost everywhere.
Here "step mappings" by Lang is the same as "simple functions" in other texts, and every such function $f_n: X\to E$ is from a measure space to a general Banach space. Also, what he means by a "Cauchy sequence" is in terms of the $L^1$-norm. That is, for every $\epsilon>0$, ther exists $N\in\mathbb{N}$ such that $\int_X |f_n-f_m|<\epsilon$ whenever $m,n>N$.
Question: Is there an example of such a sequence of simple functions $\{f_n\}$ that is Cauchy under the $L^1$-norm but does not converge pointwise outside of a measure-zero set? Or, can we push the lemma further to conclude the sequence itself must converge pointwise almost everywhere, not just a subsequence?