On the lattice structure of weakly continuous operators on the space of measures

TL;DR

This paper characterizes the lattice of weakly continuous operators on measures, showing they form a sublattice isomorphic to transition kernels, and provides an analytic proof of Doob's stability theorem.

Contribution

It establishes a lattice-theoretic structure of weakly continuous operators on measures and links them to transition kernels, offering new insights and an analytic proof of a classical result.

Findings

Weakly continuous operators form a sublattice isomorphic to transition kernels

Provides an analytic proof of Doob's stability theorem

Characterizes the lattice structure of operators on measures

Abstract

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.