  Let $G$ be a Lie group with Lie algebra $\mathfrak g$ and let $\pi$ be a
unitary representation of $G$ realized on a reproducing kernel Hilbert space.
We use Berezin quantization to study spectral measures associated with
operators $-id\pi(X)$ for $X\in {\mathfrak g}$. As an application, we show how
results about contractions of Lie group representations give rise to results on
convergence of sequences of spectral measures. We give some examples including
contractions of $SU(1,1)$ and $SU(2)$ to the Heisenberg group.
