I am trying to learn some measure theory from Lang's Real and Functional Analysis and am having difficulty understanding a claim he makes without proof.
Let $(X, \scr{M}, \mu)$ be a measured space, and let $f:X\to E\;\;$ be a function into a Banach space $E$. Suppose that $\{\varphi_n\}$ is a sequence of step maps from $X$ into $E$ which converges pointwise to $f$ almost everywhere. (Lang calls such functions $\mu$-measurable). Show that then $f$ vanishes outside some $\sigma$-finite subset of $X$.
Any help would greatly be appreciated.