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

$\blacksquare$