Pointwise a.e. Convergence

Definition

Denote our measure space as $(X,{\mathcal {M}},\mu )$.

Define the set $N=\{x:f(x)\neq \lim _{n\to \infty }f(x)\}$. A sequence of measurable functions $\{f_{n}\}$ converges to Failed to parse (unknown function "\math"): {\displaystyle f<\math>pointwise almost everywhere if <math> f_m (x) \to f(x)<\math> for almost every <math> x } , or <math> \mu(N) =0< /math>.