A Local Ratio Theorem

Abstract

Let Tt, t > 0, be a strongly continuous semigroup of positive linear contractions on the L1-space of a σ-finite measure space . We denote the integral ∫0tTsƒ ds, ƒ ∈ L1, by S0tƒ, which is denned as the limit of Riemann sums, in the norm topology of L1. It is easy to see that, given ƒ ∈ L1+, there exists a function F on the product space X× (0, ∞), measurable with respect to the usual product σ-field, such that for every t ≧ 0, ∫0tF(·, s) ds gives a representation of S0tƒ. We write S0tƒ(x) for ∫0tF(x, S) ds, with a fixed choice of F.