# Example of a set which is not in the product $\sigma$-algebra

Let $$L_d$$ be the $$\sigma$$-algebra of Lebesgue measurable subsets of $$\mathbb{R}^d$$.

By using Vitali's set $$E \subseteq [0,1]$$, I am looking for an example of $$A \in L_2$$ which is not in the product $$\sigma$$-algebra $$L_1 \times L_1$$.

I am also having trouble proving that $$L_1 \times L_1 \subseteq L_2$$. I can see that we can use $$\mathcal{B}(\mathbb{R^2})=\mathcal{B}(\mathbb{R}) \times \mathcal{B}(\mathbb{R})$$ and that the Lebesgue measure $$\lambda_2$$ on $$(\mathbb{R^2},\mathcal{B}(\mathbb{R^2}))$$ is identical to the product measure $$\lambda_1 \times \lambda_1.$$ Although I'm stuck afterwards.

1. $$L_1\otimes L_1$$ is generated by $$\mathcal{C}=\{A\times B:A,B\in L_1\}$$. Since $$\mathcal{C}\subset L_2$$, $$L_1\otimes L_1\subseteq L_2$$.

2. For a set $$N\in L_1$$ s.t. $$N\ne \emptyset$$ and $$\lambda_1(N)=0$$, the set $$E\times N\in L_2$$ ($$\because E\times N\subset [0,1]\times N$$ and $$\lambda_2([0,1]\times N )=0$$) but not in $$L_1\otimes L_1$$ ($$\because$$ for any $$L_1\otimes L_1$$-measurable set $$A$$, the sections $$A^y=\{x\in \mathbb{R}:(x,y)\in A\}$$ belong to $$L_1$$).