almost everywhere Vs. almost sure

Question

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?

Answer 1

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.

Answer 2

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.