  We introduce the matching measure of a finite graph as the uniform
distribution on the roots of the matching polynomial of the graph. We analyze
the asymptotic behavior of the matching measure for graph sequences with
bounded degree.
  A graph parameter is said to be estimable if it converges along every
Benjamini-Schramm convergent sparse graph sequence. We prove that the
normalized logarithm of the number of matchings is estimable. We also show that
the analogous statement for perfect matchings already fails for d-regular
bipartite graphs for any fixed d at least 3. The latter result relies on
analyzing the probability that a randomly chosen perfect matching contains a
particular edge.
  However, for any sequence of d-regular bipartite graphs converging to the
d-regular tree, we prove that the normalized logarithm of the number of perfect
matchings converges. This applies to random d-regular bipartite graphs. We show
that the limit equals to the exponent in Schrijver's lower bound on the number
of perfect matchings.
  Our analytic approach also yields a short proof for the Nguyen-Onak (also
Elek--Lippner) theorem saying that the matching ratio is estimable. In fact, we
prove the slightly stronger result that the independence ratio is estimable for
claw-free graphs.
