Measurability of projections Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces. Consider the product space $X\times Y$ with the product $\sigma$-algebra $\mathcal{A}\otimes \mathcal{B}$ and $\pi:X\times Y \rightarrow X$ the natural projection.
Is it true that $Z \in \mathcal{A}\otimes \mathcal{B}$ implies $\pi(Z)\in \mathcal{A}$?
(that is, the projection sends measurable sets in measurable sets)
There is a topological analogue of this results («projections are open») so I wonder if we could extend it to measure theory. 

Lebesgue thought so in the context of real line but young Souslin, famously, constructed a Borel (G-delta is enough) subset of plane with non Borel projection.
