Given sets $X$ and $K$, the projection map $\pi_{X}\colon X\times K\rightarrow X$ is defined by $\pi_{X}(x,y)=x$. An important property of analytic sets is that they are stable under projections.

The proof of this follows easily from the definition of analytic sets. First, there is a compact paved space $(K^{\prime},\mathcal{K}^{\prime})$ and a set $T\in\left(\mathcal{F}\times\mathcal{K}\times\mathcal{K}^{\prime}\right)_{{\sigma\delta}}$ such that $S=\pi_{{X\times K}}(T)$. Then,

However, $(K\times K^{\prime},\mathcal{K}\times\mathcal{K}^{\prime})$ is a compact paved space (see products of compact pavings are compact), which shows that $\pi_{X}(S)$ satisfies the definition of $\mathcal{F}$-analytic sets.

Theorem 1 above can be used to prove the following result for projections from the product of a measurable space and a Polish space. For $\sigma$-algebras $\mathcal{F}$ and $\mathcal{B}$, we use the notation $\mathcal{F}\otimes\mathcal{B}$ for the product $\sigma$-algebra, in order to distinguish it from the product paving $\mathcal{F}\times\mathcal{B}$.

An immediate consequence of this is the measurable projection theorem.

Although Theorem 2 applies to arbitrary Polish spaces, it is enough to just consider the case where $Y$ is the space of real numbers $\mathbb{R}$ with the standard topology. Indeed, all Polish spaces are Borel isomorphic to either the real numbers or a discrete subset of the reals (see Polish spaces up to Borel isomorphism), so the general case follows from this.

If $Y=\mathbb{R}$, then the Borel $\sigma$-algebra is generated by the compact paving $\mathcal{K}$ of closed and bounded intervals. The collection $a(\mathcal{F}\times\mathcal{K})$ of analytic sets is closed under countable unions and countable intersections so, by the monotone class theorem, includes the product $\sigma$-algebra $\mathcal{F}\otimes\mathcal{B}$. Then, as the analytic sets define a closure operator,

Thus every $\mathcal{F}\otimes\mathcal{B}$-analytic set is $\mathcal{F}\times\mathcal{K}$-analytic, and the result follows from Theorem 1.