Equivalence A.E.

Definition §

Let $f:\mathbb{R}\to \mathbb{R}$ be measurable. When $f=g$ a.e., we write $f\sim g$. Define $\bar{f}=\{g\mid f\sim g\}$. All such $\bar{f}$ form disjoint equivalence classes of functions that coincide a.e.

Properties §

1. (reflexivity) $f\sim f$
2. (symmetric) $f\sim g⟺g\sim f$
3. (transitive) $f\sim g,g\sim h⟹f\sim h$
4. (closed under linear combination) $f_1\sim f_2,g_1\sim g_2⟹\alpha f_1+\beta g_1\sim \alpha f_2+\beta g_2$ (where $\alpha ,\beta \in \mathbb{R}$)
5. If $f,g$ are integrable and $f\sim g$, then $\int f=\int g$.
6. If $f,g$ are continuous and $f\sim g$, then $f(x)=g(x)$ everywhere. Consequently, any equivalence class $\bar{f}$ contains at most 1 continuous function.

Remarks §

1. By convention, we do not distinguish between functions $f,g$ if $f\sim g$.