Equivalence of Definitions of Almost-Everywhere Equality Relation on Lebesgue Space
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Theorem
The following definitions of the concept of Almost-Everywhere Equality Relation on Lebesgue Space are equivalent:
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:
- $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$
Proof
Let $f, g \in \map {\LL^p} {X, \Sigma, \mu}$.
By P-Seminorm of Function Zero iff A.E. Zero, we have:
- $\norm {f - g}_p = 0$ if and only if $f - g = 0$ $\mu$-almost everywhere.
From Pointwise Addition preserves A.E. Equality, we have:
- $\norm {f - g}_p = 0$ if and only if $f = g$ $\mu$-almost everywhere.
That is:
- $\norm {f - g}_p = 0$ if and only if $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$.
$\blacksquare$