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 $n\to\infty$. Equivalently,

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 $p=1$.