An existence theorem for probability measures invariant under a Markov kernel

  • Wolfgang Adamski


LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allAAA. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.


Probability Measure Existence Theorem Vector Lattice Markov Operator Markov Kernel 
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Copyright information

© Springer 1990

Authors and Affiliations

  • Wolfgang Adamski
    • 1
  1. 1.Mathematisches Institut derUniversität MünchenMünchen 2Germany

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