Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d defined bywhere X 1 , . . . , X d are independent copies of a real-valued, centered, anisotropic Gaussian random field X 0 which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range X([0, 1] N ).We also provide a sufficient condition for a Gaussian random field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian random fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (Indiana Univ. Math.
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