Definition:Complete Measure Space

Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Suppose that the family of $\mu$-null sets $\mathcal{N}_{\mu}$ satisfies the following condition:

$\forall N \in \mathcal{N}_{\mu}: \forall M \subseteq N: M \in \mathcal{N}_{\mu}$

That is, any subset of a $\mu$-null set is again a $\mu$-null set.

Then $\left({X, \Sigma, \mu}\right)$ is said to be a complete measure space.