# show that Product Measure is not complete

Let { $\mathbb R,Μ, λ$} be the complete measure space of Lebesgue measurable subsets of $\mathbb R$, which has been constructed with the extension theorem. I got to show that the Product Measure in { $\mathbb R ^2$, $M⊗M$, $λ⊗λ$} is not complete by describing a set M ⊂ $\mathbb R ^2$ of which the Lebesgue measure is zero. Meaning that for every ε > 0 there are existing finite cuboids $Q_1$, $Q_2$,.. with M ⊂ $\cup_{n= 1}^\infty Q_n$ and $\sum _{n=1}^{\infty} 1(λ ⊗ λ)(Q_n) < ε$ that aren't $λ ⊗ λ$-measurable. I've got no idea how to work with these cuboids .. anyone who can help me?

• How about $N\times\{0\}$, where $N\subset\Bbb R$ is not Lebesgue measurable? – John Dawkins Nov 30 '15 at 20:47
• Thanks. That would be M:=N x {0} right? And the product? How it looks like? – Thesinus Dec 1 '15 at 12:22