Table Of ContentProbability and Its Applications
Radu Zaharopol
Invariant
Probabilities
of Transition
Functions
Probability and Its Applications
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Radu Zaharopol
Invariant Probabilities
of Transition Functions
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ISSN1431-7028 ProbabilityandItsApplications
ISBN978-3-319-05722-4 ISBN978-3-319-05723-1(eBook)
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Acknowledgements
It is a great pleasure to acknowledge the various kinds of support that I received
fromthefollowingpeople:EduardEmel’yanovshowedinterestinbothmymono-
graph [143] and this book by making a series of useful comments including the
improvementin the Lasota-Yorke lemma that appears in Theorem 1.2.3; Thomas
Hempflinghasshowninterestinthemonographandhasofferedmeexpertguidance
throughoutthepublicationprocess;MariusIosifescuhasalwaysbeenavailablefor
ème
extremelyusefuladvice;healsoinvitedmetothe9 ColloqueFranco-Roumain
deMath.Appl.,28Août-2Septembre2008,Bras¸ov,Romania,togiveatalkonsome
ofthetopicsdiscussedhere;MarinaReizakisandherteamatSpringerhaveexpertly
prepared the book for publication; Daniel W. Stroock brought to my attention
informationconcerningergodicmeasuresthatappearin hisbooks[119] and[29];
Tomasz Szarek sent me several of his works that I have used here; Daniël Worm
showedgreatinterestintheresultsontheKBBY decompositionthatIobtainedin
this bookand elsewhere (as I pointout several times, he has extendedthe KBBY
decomposition to Polish spaces and has obtained various other very interesting
resultsrelated to the decomposition);andlastly Marina Zaharopolhas offeredme
veryusefuladvice.
vii
Contents
1 PreliminariesonTransitionProbabilities................................. 1
1.1 BasicDefinitionsandResults.......................................... 2
1.2 InvariantProbabilities.................................................. 20
1.3 TheErgodicDecompositionofKryloff,Bogoliouboff,
BeboutoffandYosida .................................................. 25
1.4 FellerTransitionProbabilities ......................................... 36
1.4.1 SupportsofElementaryandErgodicInvariant
Measures, Minimality,Unique Ergodicity,
andGenericPoints............................................. 36
1.4.2 Equicontinuity................................................. 45
2 PreliminariesonTransitionFunctionsandTheirInvariant
Probabilities .................................................................. 57
2.1 TransitionFunctions.................................................... 57
2.2 Examples................................................................ 68
2.2.1 TransitionFunctionsDefinedbyOne-Parameter
SemigroupsorGroupsofMeasurableFunctions:
GeneralConsiderations........................................ 69
2.2.2 Transition Functions Defined by Specific
One-Parameter Semigroups or Groups
ofMeasurableFunctions...................................... 72
2.2.3 TransitionFunctionsDefinedbyOne-Parameter
ConvolutionSemigroupsofProbabilityMeasures........... 83
2.3 InvariantProbabilityMeasures ........................................ 89
3 PreliminariesonVectorIntegralsandAlmostEverywhere
Convergence.................................................................. 97
3.1 TheBochnerandtheDunford-SchwartzIntegrals.................... 97
3.1.1 TheBochnerIntegral.......................................... 98
3.1.2 CompleteMeasureSpaces .................................... 102
3.1.3 TheDunford-SchwartzIntegral............................... 105
ix
