An inverse of Furstenberg's correspondence principle and applications to nice recurrence

We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems (X,μ,T) and A⊂ X measurable there exists a set E ⊂ N such that \[ μ( ᵢ₌₁^k T-n_iA) = limN→ ∞ {|( ᵢ₌₁^k (E-n_i) )∩ \{0,,N-1\}|}{N}\] for all k,n₁,,nₖ ∈ N. As a corollary we show that a set R⊂ 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.