Let $\mathcal{L}(\mathbb{R})$ be the lebesgue-measurable set of $\mathbb{R}$, and $\mathcal{L}(\mathbb{R^2})$ the lebesgue-measurable set of $\mathbb{R^2}$.
First I shall show, that $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R})$ is a subset of $\mathcal{L}(\mathbb{R^2})$.
In a second part I shall show, that in fact they are not equal.
There is a hint, that the Lebesgue-measurable sets are the completion of the Borel sets.
So $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R})$ is the smallest $\sigma$-Algebra containing the set $\left\{ A \times B: A \in \mathcal{L}(\mathbb{R}), B \in \mathcal{L}(\mathbb{R}) \right\}$
But how to show all sets in this $\sigma$-Algebra are in fact lebesgue measurable in $\mathbb{R}^2$? I have no idea where to start. Any hint would be welcome.