Almost everywhere convergence

Almost everywhere convergence of a sequence of measurable functions means the following: a sequence $(f_n)$ of measurable functions on a measure space $(X,\Sigma,\mu)$ converges almost everywhere to a function $f$ if there exists a null set $N\subseteq X$ such that for every $x\in X\setminus N$ the limit of a sequence $f_n(x)\to f(x)$ holds as $n\to\infty$ .

Equivalent characterizations

Equivalently,

\mu\bigl(\{x\in X : f_n(x)\not\to f(x)\}\bigr)=0.

Remarks

Almost everywhere convergence is a central hypothesis in results such as the dominated convergence theorem and monotone convergence theorem. It is one of the standard modes of convergence alongside convergence in measure and convergence in Lp.

Examples

On $([0,1],\mathcal{B},\lambda)$ , the sequence $f_n(x)=x^n$ converges almost everywhere to $f(x)=0$ (the only exceptional point is $x=1$ ).
On $((0,1),\mathcal{B},\lambda)$ , the functions $f_n(x)=n\,\mathbf{1}_{(0,1/n)}(x)$ satisfy $f_n(x)\to 0$ almost everywhere, but $\|f_n\|_1=\int_0^1 f_n\,d\lambda=1$ for all $n$ , so the convergence is not convergence in Lp when $p=1$ .