Almost-Everywhere Equality Relation for Measurable Sets is Equivalence Relation

Theorem

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.

Then:

$\sim_\mu$ is an equivalence relation.

Proof

Checking in turn each of the criteria for equivalence:

Reflexivity

By Symmetric Difference with Self is Empty Set, we have:

We have:

$A \symdif A = \O$

By Measure of Empty Set is Zero:

$\map \mu \O = 0$

So:

$\map \mu {A \symdif A} = 0$

Therefore:

$A \sim_\mu A$

Hence $\sim_\mu$ is a reflexive relation.

$\Box$

Symmetry

Suppose:

$A \sim_\mu B$

Then:

$\map \mu {A \symdif B} = 0$

We have:

$\map \mu {B \symdif A} = 0$

Therefore:

$B \sim_\mu A$

Hence $\sim_\mu$ is a symmetric relation.

$\Box$

Transitivity

Suppose:

$A \sim_\mu B$

and:

$B \sim_\mu C$

Then:

$\map \mu {A \symdif B} = 0$

And:

$\map \mu {B \symdif C} = 0$

Therefore:

$\map \mu {A \symdif C} = 0$

Hence $\sim_\mu$ is a transitive relation.

$\Box$

All the criteria are therefore seen to hold for $\sim_\mu$ to be an equivalence relation.

$\blacksquare$

Sources

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

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