Space of functions on $[a, b]$ with countable discontinuity We are quite familiar with $C([a,b])$, the set of all continuous functions defined on a closed interval $[a,b]$ with the supremum norm. My question is; if we allow function with countable discontinuity what will change? I think everything will change but I do not know how to approach this problem. As an example if I allow countable discontinuity is the usual norm valid in $C([a,b])$ is applicable? What condition we need to impose to make the space complete? Etc. Is this space studied in depth? I could not figure out much in the literature. Any suggestion?
 A: Here is one particular example:
If we consider Lebesgue measure, then we can consider $L^\infty([a,b],m)$, which has a norm given by $$\|\cdot\|_\infty \stackrel{\textrm{def}}{=} \operatorname{ess} \sup_{[a,b]} \cdot.$$
The essential supremum is defined as the smallest upper bound $a$ of $f$ such that $m\left(\{x : f(x) > a\}\right)) = 0$. This space is complete with respect to the topology induced by the $\infty$-norm provided that our measure space is semi-finite. Below is a (probably ham-handed) outline of the proof of completeness.
Let us take an absolutely convergent series in $L^\infty([a,b],m)$ converging to $f$. Then $\sum_1^\infty \|f_k\|_\infty = B < \infty$. For any $\epsilon > 0$, consider a set $B_\epsilon = \{ x : f(x) > \|f\|_\infty - \epsilon\}$. Now, take a finite-measure subset $A$ of $B_\epsilon$.
On $A$, we have $|f_k| < \|f_k\|_\infty$ by definition, so we must have $\sum_1^\infty |f_k|$ converging, and in fact, the convergence is uniform. Thus, we must have $|f_k - f| \to 0$ on any set with non-zero finite measure. In a normed vector space, we have that the space is complete iff every absolutely convergent series converges in the space, so the space described above is in fact complete.
