  We prove an inverse of Furstenberg's correspondence principle stating that
for all measure preserving systems $(X,\mu,T)$ and $A\subset X$ measurable
there exists a set $E \subset \mathbb{N}$ such that \[ \mu\left(
\bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left(
\bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] for all
$k,n_1,\dots,n_k \in \mathbb{N}$. As a corollary we show that a set $R\subset
\mathbb{N}$ is a set of nice recurrence if and only if it is nicely
intersective. Together, the inverse of Furstenberg's correspondence principle
and it's corollary partially answer two questions of Moreira.
