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Let X(t) be a Gaussian random field R d $\rightarrow$ R. Using the notion of(d -- 1)-integral geometric measures, we establish a relation between (a) thevolume of the level set (b) the number of crossings of the restriction of therandom field to a line. Using this relation we prove the equivalence betweenthe finiteness of the expectation and the finiteness of the second spectralmoment matrix. Sufficient conditions for finiteness of higher moments are alsoestablished.

Conditions for the finiteness of the moments of the volume of level sets