Definition:Almost-Everywhere Equality Relation/Measurable Functions/Real-Valued Functions
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Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.
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:
- $\map \mu {\set {x \in X : \map f x \ne \map g x} } = 0$