Let $X(t)$ be a Gaussian random field $\mathbb R^d\to \mathbb R$. Using the notion of $(d-1)$-integral geometric measures, we establish a relation between (a) the volume of level sets, and (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.
"Conditions for the finiteness of the moments of the volume of level sets." Electron. Commun. Probab. 24 1 - 8, 2019. https://doi.org/10.1214/19-ECP214