Let $F$ be a set and $\mathcal F$ a collection of subsets of $F$ such that $\emptyset \in \mathcal F$. We denote by $F_\sigma$ (resp. $F_\delta$) the closure of $F$ under countable union (resp. intersection). Let $F_{\sigma \delta} := (F_\sigma)_\delta$. If $E$ is a topological space, then $\mathcal K (E)$ denotes the collection of all compact subsets of $E$.
Let $F$ be a metric space together with its Borel $\sigma$-algebra $\mathcal F$. A subset $A$ of $F$ is called $\mathcal F$-analytic if
- Def 1 there exists a Polish space $E$ and a continuous map $f:E \to F$ such that $A = f(E)$.
- Def 2 there exists a compact metric space $E$ and $B \in (\mathcal K (E) \times \mathcal F)_{\sigma \delta}$, such that $A$ is the projection of $B$ onto $F$.
Def 2 is taken from page 41 of Dellacherie/Meyer's Probabilities and Potential.
It is mentioned here that they are equivalent, but there is no proof. Could you elaborate on how to prove this equivalence?