Definition:Almost-Everywhere Equality Relation

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Measurable Functions

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

Real-Valued Functions

Let $\map {\mathcal M} {X, \Sigma, \R}$ be the real-valued $\Sigma$-measurable functions on $X$.


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

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

That is:

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


Extended Real-Valued Functions

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


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

$f \sim_\mu g$ if and only if $\map f x = \map g x$ 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:

$f \sim_\mu g$ if and only if $\norm {f - g}_p = 0$

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:

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

That is:

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



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