Almost everywhere

A property $P(x)$ is said to hold almost everywhere (with respect to a measure $\mu$ on $X$) if there exists a null set $N\subseteq X$ such that $P(x)$ holds for all $x\in X\setminus N$.

This notion depends on the underlying measure space and is weaker than pointwise validity: changing a function on a null set does not affect almost-everywhere statements. It is also closely tied to the symmetric difference of sets.

Examples:

• Two functions $f,g:X\to\mathbb R$ are equal almost everywhere if $\{x\in X: f(x)\ne g(x)\}$ is a null set.
• For measurable sets $A,B\in\Sigma$, the indicator functions $\mathbf 1_A$ and $\mathbf 1_B$ are equal almost everywhere exactly when $A\triangle B$ is a null set.