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?