A Decomposition of Excessive Measures

  • R. M. Blumenthal
Part of the Progress in Probability and Statistics book series (PRPR, volume 12)


Let {Pt;t ≥ 0} denote the transition semigroup for a Borel right Markov process on a state space (E, ℰ). We set \( U\left( {x,D} \right) = \int\limits_0^{\infty } {{P_t}} \left( {x,D} \right)dt \) and denote by µU and Uf the potential of a measure μ and a function f respectively. Given a measure m on ℰ and a set D in ℰ let mD be the measure mD(B) = m(D∩B). A measure m on ℰ is called excessive if it is σ-finite and m ≥ mPt for all t. If µ is a measure on ℰ the formula \( \mu U(A) = \int {\mu \left( {dx} \right)U\left( {x,A} \right)} \) defines a measure, the potential of µ, on ℰ. It will be excessive if and only if it is σ-finite. An excessive measure m is called invariant if mPt = m for all t.


Markov Process Borel Function Finite Measure Countable Union Invariance Result 
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Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • R. M. Blumenthal
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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