Almost Everywhere Equivalence Relation

Last updated Nov 1, 2022

Let $(X, \mathcal{M}, \mu)$ and $(Y, \mathcal{N}, \nu)$ be Measure Spaces. Define the relation $\sim$ on $\mathcal{L}^{0}$ as $f \sim g$ for $f,g \in \mathcal{L}^{0}$ if ${x \in X : f(x) \neq g(x)}$ is a $\mu$-Null Set. Then $\sim$ is an Equivalence Relation.

We define $L^{0} := \mathcal{L}^{0} / \sim$.

1. If $f \in \mathcal{L}^{0}$, then ${x \in X : f(x) \neq f(x)} = \emptyset$. Since $\mu(\emptyset) = 0$, $f \sim f$. $\checkmark$
2. If $f,g \in \mathcal{L}^{0}$, then ${x \in X : f(x) \neq g(x)} = {x \in X : g(x) \neq f(x)}$. Thus $f \sim g \Rightarrow g \sim f$. $\checkmark$
3. Suppose $f,g,h \in \mathcal{L}^{0}$ and $f \sim g$ and $g \sim h$. Then ${x \in X : f(x) \neq g(x)}$ and ${x \in X : g(x) \neq h(x)}$ are both Null Sets. Note that if for $x \in X$, we have $f(x) \neq h(x)$, then we must have one of $f(x) \neq g(x)$ or $g(x) \neq h(x)$. Otherwise $f(x) = g(x) = h(x)$. Thus $${x \in X : f(x) \neq h(x)} \subset {x \in X : f(x) \neq g(x)} \cup {x \in X : g(x) \neq h(x)}.$$ Since Counable Union of Null Sets is Null and by Monotonicity of Measures, we see $f \sim h$. $\checkmark$ $\blacksquare$