  We associate to a test configuration of an ample line bundle a filtration of
the section ring of the line bundle. Using the recent work of Boucksom-Chen we
get a concave function on the Okounkov body whose law with respect to Lebesgue
measure determines the asymptotic distribution of the weights of the test
configuration. We show that this is a generalization of a well-known result in
toric geometry. As an application, we prove that the pushforward of the
Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure
of a certain deformation of the manifold. Via the Duisteraat-Heckman formula,
we get as a corollary that in the special case of an effective
$\mathbb{C^{\times}}$-action on the manifold lifting to the line bundle, the
pushforward of the Lebesgue measure on the Okounkov body is piecewise
polynomial.
