Complete Measure Spaces

# Complete Measure Spaces

 Definition: A complete measure space $(X, \mathcal A, \mu)$ is Complete if for every measurable set $E \in \mathcal A$ with $\mu (E) = 0$ we have that every subset of $E$ is a measurable set. If $(X, \mathcal A)$ is not complete we say it is Incomplete.

It is important to note the distinction between a complete measure space and a complete metric space. These two concepts are NOT the same.

Consider the Lebesgue measure space $(\mathbb{R}, \mathcal M, m)$. Let $E$ be a Lebesgue measurable set and let $E' \subseteq E$. Then by the monotonicity of the Lebesgue outer measure we have that:

(1)
\begin{align} \quad m^*(E') \leq m^*(E) = m(E) = 0 \end{align}

Therefore $m^*(E') = 0$. From one of the results on the Lebesgue Measurable Sets page, since $m^*(E') = 0$ we have that $E'$ is a Lebesgue measurable set. Hence, every subset of a Lebesgue measurable set $E$ with $m(E) = 0$ is also a Lebesgue measurable set. Therefore $(\mathbb{R}, \mathcal M, m)$ is a complete measure space.

For another example, consider the counting measure space $(X, \mathcal P(X), c)$ and suppose that for $E \in \mathcal P(X)$ we have that $c(E) = 0$. Then $E = \emptyset$. The only subset of $E$ is $\emptyset$ itself and so $(X, \mathcal P(X), c)$ is trivially a complete measure space.