Shift-generated classes of jointly measurable random fields

We study shift-generated classes of jointly measurable and separable \(\mathbb R^d\)-valued random fields (RFs) indexed by \(\mathbb R^l\), defined through identities for \(\alpha\)-homogeneous functionals.

In contrast to earlier work, no stochastic-continuity assumption and no local boundedness condition are imposed.

We show that every non-empty shift-generated class contains an \(L^\alpha\)-continuous element.

This regularization result allows us to establish the strict positivity of the integral functional for all elements of the class and for the associated local RFs.

We further extend the defining functional identity to a larger class of functionals, including integral functionals, and use this to construct canonical elements of a given class via randomised shifts.

We also relate shift-generated classes to spectral tail and tail RFs and show that every spectral tail RF has an \(L^\alpha\)-continuous representative with the same finite-dimensional distributions.

As an application, we identify the \(-\alpha\)-homogeneous tail measure associated with a shift-generated class and show that it depends only on the class and admits an \(L^\alpha\)-continuous representor.