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Is there an abbreviation for "almost all $x\in X$?

I have "$\forall a.e. x\in X$" in my mind, but i see nobody uses this..

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Although the standard is "a.e", I've heard that some people used to use "p.p.", which stands for the French "presque partout" - "almost everywhere". – Isaac Solomon Jan 15 '13 at 5:03
I would find $\forall a.e.$ confusing as $\forall$ by itself denotes "for all." Hence $\forall a.e.$ would seem to denote "for all almost every." I think a.e. is a good abbreviation, and artificially defining another abbreviation seems not worth the trouble. Writing "For almost every" is painless and avoids abbreviations (and hence any confusion). – JavaMan Jan 15 '13 at 5:09
Not attempting to answer the question, but commenting on-topic: I've always liked the visual information quickly conveyed by $\forall$ (something the phrase "for all" just can't do), so in my own notes I've started using $\stackrel{a.}{\forall}$ – Dahn Jahn Jan 15 '13 at 9:08
I myself have used: "for a.a. $x\in X$" or "for $\mu$-a.a. $x\in X$" to emphasize which measure it is with respect to. – Stefan Hansen Jan 15 '13 at 9:18
You could write $\forall ^\mu x \in X,$ or even $\forall x \in (X,\mu)$. Arguably, the "correct" definition of universal quantification in a measure space is the a.e. version. – goblin Jun 21 '14 at 8:08

4 Answers 4

up vote 8 down vote accepted

You can use $\mu$-a.e $x\in X$ (because depends on measure).

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I write this as a.e. $[\mu]$. – mrf Jan 15 '13 at 9:02
And probabilists write $\mu$-a.s. (almost surely). – Michael Greinecker Jan 15 '13 at 9:31

It's common to see things like if $\int_a^b |f|dx=0$ then for a.e. $x \in [a,b]$ we have $f(x)=0$. I've never seen the notation $\forall a.e. x \in X$ personally. Of course it's also not too many characters to write out for almost every $x \in X$.

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"we say P(x) for a.a. x in X, standing for almost all x in X."

Of course, this is just something somebody said, not a well-accepted textbook. There is however, this very official-looking index for an unknown math text on MIT Press's website:

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The book RussH linked to says $\dot\forall x \in X$, written \dot\forall x in TeX math mode.

'Optimal Control Theory with Applications in Economics', by Weber, Appendix A. Thanks RussH!

I'm coming to this question with a density $f(\cdot)$, so I don't want to use $\mu$-a.e. $x \in X$. That would require stating that the measure induced by the density is called $\mu(\cdot)$. But $\dot{\forall}x$ is (by default) with respect to Lebesgue measure, which works for my purposes. Using words also doesn't really fit well into e.g. the LaTeX align environment.

edit: I guess one could also introduce $\mu$ as Lebesgue measure, which would amount to the same thing.

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