Product of complete measures Suppose $(X, \mathcal M, \mu)$ and $(Y, \mathcal N, \nu)$ are two measure spaces, prove $\overline {\mathcal M \otimes \mathcal N}= \bar{\mathcal M} \otimes \bar {\mathcal N}$.
Honestly, I don't have any ideas about this one. I only know the definition that the subset of every null set is measurable, but don't know how to apply to the product of complete measures. 

Let $X = Y = \mathbb{R}$ and $\mathcal{M} = \mathcal{N} = \mathcal{L}$, the Lebesgue measurable sets. Let $\mu = \nu = \lambda$, the Lebesgue measure.
Suppose that $S \in \mathcal{M}$ is $\mu$-null. Since $\mathcal{N} \neq \mathcal{P}(Y)$, there exists some $T \in \mathcal{P}(Y) \setminus \mathcal{N}$. Then it can be shown that $S \times T \not\in \mathcal{M} \times \mathcal{N} = \mathcal{L}^2$, but it is contained in $S \times Y$, which has $0$ product measure.
So the product of two complete measures may not be complete.
