Convergence in $L^p$

A convergence in $L^p$ is norm convergence in a Lp space. Let $(X,\mathcal{A},\mu)$ be a measure space and let $1\le p<\infty$. A sequence $(f_n)$ in $L^p(X)$ converges in $L^p$ to $f\in L^p(X)$ if

where $\|\cdot\|_p$ is the Lp norm. For $p=\infty$, one defines convergence in $L^\infty$ by $\|f_n-f\|_\infty\to 0$, where $\|\cdot\|_\infty$ is the essential supremum norm (see essential supremum).

Convergence in $L^p$ controls the size of the error in an averaged sense. In particular, for $1\le p<\infty$ the estimate

shows that convergence in $L^p$ implies convergence in measure.

for all $n$.