  We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom.
\textbf{18} (2009), no. 2, 279--308); Boucksom and H. Chen (Compos. Math.
\textbf{147} (2011), no. 4, 1205--1229); and S. Boucksom, A. K\"{u}ronya, C.
Maclean, and T. Szemberg (Math. Ann. \textbf{361} (2015), no.~3--4, 811--834).
The key point is to associate a Duistermaat-Heckman measure to a filtered big
linear series on a given projective variety. The expectation of the measure
admits a description via the theory of Newton-Okounkov bodies. Such
considerations have origins in symplectic geometry. They have applications for
$\mathrm{K}$-stability and Diophantine arithmetic geometry of projective
varieties.
