Let $(X, \mathcal{F}, \mu)$ be a measure space. Define

\begin{align*} \overline{\mathcal{F}} := \{ A \cup N : A \in \mathcal{F}, \; N \subset E \text{ for some } E \in \mathcal{F} \text{ with } \mu(E) = 0 \}. \end{align*}

Then $\overline{\mathcal{F}}$ is a $\sigma$-algebra, and the function $\overline{\mu}: \overline{\mathcal{F}} \to [0, \infty]$ defined by $\overline{\mu}(A \cup N) := \mu(A)$ is a well-defined complete measure extending $\mu$. The triple $(X, \overline{\mathcal{F}}, \overline{\mu})$ is called the **completion** of $(X, \mathcal{F}, \mu)$.