Now we come to one of the most common terms of art in analysis: “almost everywhere”. It’s unusual in that it sounds perfectly colloquial, and yet it has a very technical meaning.
The roots of “almost everywhere” are in the notion of a negligible set. If we’re working with a measure space we don’t really care about subsets of sets of measure zero, and anything that happens only on such a negligible set we try to sweep under the rug. For example, let’s say we have a function defined by for all , and by . Colloquially, we say that is zero “almost everywhere” because the set where it isn’t zero — the singleton — has measure zero.
In general, if we have some property that can be applied to points , then we say is true “almost everywhere” if the set where is false is negligible. That is, if we can find some measurable set with so that is true for all . Note that we don’t particularly care if the set where is false is itself measurable, although if is complete then all -negligible sets will be measurable. This sort of language is so common in measure theory and analysis that it’s often abbreviated as “a.e.”. Older texts will say “p.p.” for the French equivalent “presque partout“. In probability theory (measure theory’s cousin) we run into “a.s.” for “almost surely”.
No matter how we say or write it, “almost everywhere” has a hidden dependence on some measure. In many cases, the measure is obvious from context, in that there’s only one measure under consideration on a given space. However, in the case where we have two measures and on the same measurable space, we may distinguish them by writing “-almost everywhere” and “-almost everywhere” (or “-a.e.” and “-a.e.”), or by explicitly stating with respect to which measure we mean.
We’ve actually seen this sort of thing in the wild before; Lebesgue’s condition can be reformulated to say that a bounded function defined on an -dimensional interval is Riemann integrable on that interval if and only if is continuous almost everywhere (with respect to Lebesgue measure).
As more of a new example, we say that a function is “essentially bounded” if it is bounded almost everywhere. That is, if there is a constant and some measurable set with so that for all . We’re willing to accept some points exceeding , but no more than a set of measure zero. The infimum of all such essential bounds is the “essential supremum” of , written .