TITLE: Why aren't probability measures required to use $2^X$ as the underlying $\sigma$-algebra?
QUESTION [6 upvotes]: Let $Y$ be a random variable that takes values in some set $X$ according to a probability measure $\mu$.  If the $\sigma$-algebra on which $\mu$ is defined is not $2^X$, then there exists $A \subset X$ with $\mu(A)$ undefined.  This implies that the event "A realization $y$ of $Y$ satisfies $y \in A$" has undefined probability.  But that can't be right: if we sample $Y$ over and over, the frequency with which our event comes true should converge to some value, so the event does have a probability.
Must all probability measures be defined on $2^X$?  Or is my intuition that in the real world, all events have a probability wrong?

REPLY [4 votes]: There are several reasons why we might work with $\sigma$-algebras smaller than $2^X$:

We might want to draw a point from the unit interval $[0,1]$ in a way that no single point is drawn with positive probability. It is consistent with the usual axioms of set theory that this is impossible. In particular, it follows from a result of Ulam that this is impossible under the continuum hypothesis. 
Even stronger: We might want to have a uniform probability distribution on $[0,1]$. The Vitali construction shows that this is impossible.
We are only interested in a small class of event in practice and want efficient ways to determine their information. For the Borel $\sigma$-algebra on $\mathbb{R}$, all probability distributions are determined by a cumulative distribution function and this is quite useful in practice.

There are also problems with the relative frequency interpretation. Suppose $X=\mathbb{N}$ and nature draws the sequence $1,2,3,\ldots$. Each number gets assigned zero probability, so the relative frequencies violate countable additivity.