Equivalence of Definitions of Almost-Everywhere Equality Relation on Lebesgue Space

Proof
Let $f, g \in \map {\LL^p} {X, \Sigma, \mu}$.

By P-Seminorm of Function Zero iff A.E. Zero, we have:


 * $\norm {f - g}_p = 0$ $f - g = 0$ $\mu$-almost everywhere.

From Pointwise Addition preserves A.E. Equality, we have:


 * $\norm {f - g}_p = 0$ $f = g$ $\mu$-almost everywhere.

That is:


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