Definition:Almost-Everywhere Equality Relation/Measurable Sets

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\Sigma$ by:

$A \sim_\mu B$ if and only if $\map \mu {A \symdif B} = 0$

where $\symdif$ denotes set symmetric difference.

Sources

• 1982: Peter Walters: An Introduction to Ergodic Theory Chapter $2$: Isomorphism, Conjugacy, and Spectral Isomorphism