Uniformly contractive semigroups and submarkovian semigroups (I)

Let (M, ℱ,d, x) be a σ-finite measure space with a σ-field ℱ countably generated. We call a linear mapT uniformly contractive if which maps measurable functions onM to measurable functions and If a linear mapT which maps measurable functions onM to measurable functions has positivity property, namely,Tf≧0 forf≧0, we call it a submarkovian operator. In this article we prove