Definition:Almost-Everywhere Equality Relation

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

Let $\map {\mathcal M} {X, \Sigma}$ be the set of $\Sigma$-measurable functions on $X$.

We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\map {\mathcal M} {X, \Sigma}$ by:


 * $f \sim_\mu g$ $\map f x = \map g x$ for $\mu$-almost all $x \in X$.

That is:


 * $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$

Lebesgue Space
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.

Let $\map {\mathcal L^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.

Definition 1
We define the $\mu$-almost-everywhere equality relation $\sim_\mu$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:


 * $f \sim_\mu g$ $\norm {f - g}_p = 0$

where $\norm \cdot_p$ is the $p$-seminorm.

Definition 2
We define the $\mu$-almost-everywhere equality relation $\sim$ on $\map {\mathcal L^p} {X, \Sigma, \mu}$ by:


 * $f \sim_\mu g$ $\map f x = \map g x$ for $\mu$-almost all $x \in X$.

That is:


 * $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$

Also see

 * Almost-Everywhere Equality Relation is Equivalence Relation