Let $(X,\mu)$ be a measure space. If $\Phi\colon X\to \mathbb{R}$ is a nonnegative function with $\int \Phi d\mu <\infty$ and if $f_1, f_2,\dots$ is a sequence of measurable functions such that $|f_n|\leq \Phi$ for each $n$ then $$g=\liminf_{n\rightarrow\infty} f_n \;\;\textnormal{and}\; h=\limsup_{n\rightarrow\infty} f_n$$ are both integrable, and $$-\infty < \int g d\mu\leq \liminf_{n\rightarrow\infty}\int f_nd\mu\leq \limsup_{k\rightarrow\infty}\int f_n d\mu\leq \int h d\mu < \infty.$$