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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)?

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  • $\begingroup$ "almost everywhere $[\mu]$" is standard terminology. $\endgroup$ – Daniel Fischer Nov 29 '13 at 14:32
  • $\begingroup$ I usually say $\mu$ almost everywhere or short $\mu$-a.e. When $\mu$ is a probability measure then you would say $\mu$ almost surely or $\mu$-a.s. $\endgroup$ – Stefan Hansen Nov 29 '13 at 14:44
  • $\begingroup$ Use "almost surely" only in the context of probability theory. I agree that "almost everywhere" should be preferred to "almost all" to reduce confusion. And those other branches of mathematics that use "almost all" should likewise find alternate formulations to avoid confusion with measure theory! $\endgroup$ – GEdgar Nov 29 '13 at 14:50
  • $\begingroup$ Maybe my question was not so clear, I didn't mean the property holds in all $X$ (the whole space) except in a zero measure subset, but that it holds in all points of a subset $E$ of $X$ except in a zero measure subset of it. Would $\mu$-almost everywhere on $E$ be standard terminology in this case? $\endgroup$ – Dimas Nov 29 '13 at 18:01
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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.

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