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I'm reading a book about measure theory and probability (first chapter of Durret's Probability book), and it's starting to switch between the terms "a.e." and "a.s." in different contexts. I'm becoming confused about their meanings. What's the difference between almost everywhere and almost sure?

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    $\begingroup$ a.e. is used for general measure spaces, while a.s. is used for probabilistic spaces. There is no important difference between both concepts (for example, measurable functions are named random variables when the measure space is a probabilistic space). $\endgroup$
    – sinbadh
    May 12, 2016 at 8:52
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    $\begingroup$ a.e. can be used a.e. This in contrast with a.s. which is only used in the context of probability measures. In principle you could use a.e. there too, but a.s. a.e. is replaced there by a.s. $\endgroup$
    – drhab
    May 12, 2016 at 9:17

3 Answers 3

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In a probability space (equipped with a probability $P$), we say that an event $\omega$ occurs almost surely if $P(\omega)=1$. On the other hand, on a measure space equipped with a measure $\mu$, we say that a property $\mathcal{P}$ is satisfied almost everywhere if the set where $\mathcal{P}$ is not satisfied has measure zero. Note that "a.s." is equivalent to "a.e." in probability spaces, since if $\omega$ occurs almost surely, then the probability that $\omega$ does not occur is zero. However, in the case of general measure spaces $X$ we cannot say that a property is satisfied almost everywhere if it is satisfied in a set of measure $\mu(X)$ (which would correspond to an event having probability $1$), since in many cases this measure is infinite. This is why in the case of measure spaces we formulate the definition of "almost everywhere" in terms of complements of sets.

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  • $\begingroup$ $\{x: x \in (-\infty, 0)\cup (1, +\infty)\}$ is a.s. in $\mathbb{R}$ but not a.e. in $\mathbb{R}$, $\mathbb{R}$ is equipped with Borel sets and Lebesgue measure. Is this right? $\endgroup$
    – Animeta
    Dec 12, 2019 at 16:39
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When we say that something happens "almost everywhere", we mean to say that:

  1. This something can happen or fail to happen in any point in a given measure space; and
  2. The set of points in which it fails to happen is a set of measure zero.

Notice that there's no notion of probability when talking about "almost everywhere".

Now, when we say that something happens "almost surely", we mean to say that:

  1. This something is the result of a random experiment. It can happen or fail to happen, and the result of this experiment (success ofr failure) is a random variable; and
  2. The probability of this something failing to happen is zero.
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From Wiki

Almost surely In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. In other words, the set of possible exceptions may be non-empty, but it has probability.

this means almost surely comes from probability trials (stochastic trials),

From Wolfram mathworld

Almost Everywhere

A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero.

From this book

"Essentials of Probability Theory for Statisticians"

Par Michael A. Proschan,Pamela A. Shaw page 102

Almost surely for random variable, (probability)
Almost everywhere for the sequance of functions, (measure)

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