Let $(E,\mathcal{E},\mu)$ be a [measure space](/page/Measure%20Space) such that $E\neq\varnothing$ and $\mu(E)<\infty$. Let $B(E)$ denote the real [vector space](/page/Vector%20Space) of bounded functions $g:E\to\mathbb{R}$. Equip $B(E)$ with the [uniform norm](/page/Uniform%20Norm) $\|\cdot\|_\infty:B(E)\to[0,\infty)$ defined by

\begin{align*} \|g\|_\infty=\sup_{x\in E}|g(x)| \end{align*}

for every $g\in B(E)$. Let $(f_n)_{n=1}^{\infty}$ be a sequence in $B(E)$ such that each $f_n:E\to\mathbb{R}$ is $\mathcal{E}$-measurable, and let $f\in B(E)$ be $\mathcal{E}$-measurable. If

\begin{align*} \|f_n-f\|_\infty\to 0, \end{align*}

then

\begin{align*} \int_E |f_n-f|\,d\mu(x)\to 0. \end{align*}