  We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial
measures on $\mathbb{R}^2$. These measures are defined in terms of suitable
area, dinv, and bounce statistics on continuous families of paths in the plane,
and have many combinatorial similarities to the $q,t$-Catalan numbers. Our main
result realizes the $q,t$-Catalan measures as a limit of higher $q,t$-Catalan
numbers $C^{(m)}_n(q,t)$ as $m\to\infty$. We also give a geometric
interpretation of the $q,t$-Catalan measures. They are the Duistermaat-Heckman
measures of the punctual Hilbert schemes parametrizing subschemes of
$\mathbb{C}^2$ supported at the origin.
