Let $(E,\mathcal{E},\mu)$ be a measure space, let $(f_n)_{n=1}^{\infty}$ be a sequence of $\mathcal{E}$-measurable functions $f_n:E\to\mathbb{R}$, and let $f:E\to\mathbb{R}$ be an $\mathcal{E}$-measurable function.

1. If for every $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $|f_n(x)-f(x)|<\varepsilon$ for every $n\ge N$ and every $x\in E$, then $f_n\to f$ pointwise on $E$. 2. Let $1\le p<\infty$. If $f_n\in L^p(E,\mathcal{E},\mu)$ for every $n\in\mathbb{N}$, $f\in L^p(E,\mathcal{E},\mu)$, and $\|f_n-f\|_{L^p(E,\mathcal{E},\mu)}\to 0$, then for every $\varepsilon>0$, $\mu(\{x\in E: |f_n(x)-f(x)|>\varepsilon\})\to 0$. 3. If $\mu(E)<\infty$ and there exists a measurable set $Z\in\mathcal{E}$ with $\mu(Z)=0$ such that $f_n(x)\to f(x)$ for every $x\in E\setminus Z$, then for every $\varepsilon>0$, $\mu(\{x\in E: |f_n(x)-f(x)|>\varepsilon\})\to 0$.