Integral representation of invariant functions (Q1286331)

The authors characterize functions invariant for a bounded kernel \(\Pi\) which have a Choquet-type integral representation. Such representation results are well-known if \(\Pi\) satisfies some strong regularity assumptions, while in general such an integral representation does not hold. Essential for representing an invariant function \(f\) is the existence of a \(\sigma\)-finite measure \(m\) with \(\mathbf{1}_{f<\infty} \lim_{n\to \infty}(\Pi)_{\perp m}^n f=0\) where \((\Pi)_{\perp m}^n\) is the singular part of the Lebesgue decomposition of \(\Pi^n\) with respect to \(m\). The results are applied to convolution kernels with spread-out measures on second countable locally compact Abelian groups.