We define product measure on complete measurable spaces because if $N$ is a nul set and $M$ is measurable, we want that $$m_2(N\times M)=m_1(N)m_1(M)=0.$$ Let $(\mathbb R^2, \mathcal M\times \mathcal M, m_2)$ the Lebesgue measure space on $\mathbb R^2$. We have that $$\mathcal M\times \mathcal M=\sigma (\{A\times B\mid A,B\in \mathcal M\}),$$ and it's a complete measurable space. I know that for all $A\times B\in \mathcal M\times \mathcal M$ we have that $$m_2(A\times B)=m_1(A)m_1(B).\tag{*}$$
But if $\mathcal N$ is non measurable (for example Vitali set), then $\{0\}\times \mathcal N$ has measure $0$ and thus is in $\mathcal M\times \mathcal M$. But don't we have that $$m_2(\{0\}\times \mathcal N)\neq m_1(\{0\})m_1(\mathcal N) \ \ ?$$ So $(*)$ is wrong ? Could someone explain ?