  We apply the integral formula of volumes to the family of graded linear
series constructed from any test configuration. This solves the conjecture
raised by Witt--Nystr\"{o}m so that the sequence of spectral measures for the
induced $\mathbb{C}^*$-action on the central fiber converges to the canonical
Duistermatt--Heckman measure defined by the associated weak geodesic ray. As a
consequence, we show that the algebraic $p$-norm of the test configuration
equals to the $L^p$-norm of tangent vectors. Using this result, We may give a
natural energy theoretic explanation for the lower bound estimate on the Calabi
functional by Donaldson and prove the analogous result for the
K\"{a}hler--Einstein metric.
