Let $X$ be a measure space and $E$ be a Banach space. A step function from $X$ to $E$ is a measurable function ($E$ takes the Borel measure) with finite image and whose support has finite measure.
If $(f_n)$ is a Cauchy sequence of step functions that converge a.e. to a function $f$, then do they also converge to $f$ in the $L^1$ seminorm?
The context for this question is that Lang seems to make this leap in his proof of Theorem 3.4 on page 133 of Real and Functional Analysis, and I want to know if the proof is still correct.