Let $(X,\mathcal{F},\mu)$ be a measure space. I want to know the correct jargon to say that a property holds everywhere except possibly a measure zero subset of a given set $E\in\mathcal{F}$, that is, there exists some $N\in\mathcal{F}$ with $\mu(N)=0$ such that a property holds for all $x\in E\backslash N$.
My first guess would be to say that the property holds for almost all $x\in E$, but wikipedia says that almost all can have other meanings as well, which could confuse readers. My second guess would then be saying that the property holds almost everywhere on $E$, but that sounds kind of funny (linguistically-wise). Does anyone have a suggestion? Also, what would be the probability space equivalent, saying that the property holds almost surely given $E$ (if $E$ has non-null probability)?
In the context of measure theory, "almost all" quite non-confusingly stands for "all but a set of measure zero" (esp. when dealing with a fixed measure, otherwise see Daniel Fisher's comment for a less ambiguous variant); all but finitely many and other interpretations are customary only as long as one has not mentioned measures. And you are right: "almost everywhere" is the proper versions of the same notion when the underlying topic is considered as something "spacial", and similarly "almost surely" in a probability space.