On the limit of spectral measures associated to a test configuration

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.