Suppose that $X$ is a subset of a Hausdorff topological space, and $Z\subseteq X$ a member of the family of Borel subsets on $X$. Let $B(Z)$ the set of bounded functions on $Z$, equipped with the $\sup$ norm. Then we can easily prove that every Cauchy sequence in $B(Z)$ converges.
Now take $BM(Z)$ the set of real-valued, bounded, measurable functions defined on $Z$. The fact that the functions are measurable, gives a hint as to define a norm on this space that uses this measure. Indeed, suppose that $\mu$ such a (finite, signed) measure on $Z$; the essential supremum norm $\|f\|^*_\infty = \mathrm{ess}\sup |f|$ yields the equivalence class of measurable functions that are bounded $\mu-$almost everywhere. Call this space $\mathcal{L}_\infty(Z,B(Z),\mu)$.
Question:
Suppose that we take a function $f\in \mathcal{L}_\infty$. Then this function may or may not in general exist in $B(Z)$, for it may assume an infinite value at some point in $Z$. Then how can one be assured that $\mathcal{L}_\infty$ is a vector space?