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A. BOBROWSKI
Preface pagexi
1 Preliminaries,notationsandconventions 1
1.1 Elementsoftopology 1
1.2 Measuretheory 3
1.3 Functions of boundedvariation. Riemann–Stieltjesintegral 17
1.4 Sequences of independentrandomvariables 23
1.5 Convexfunctions. H¨olderandMinkowskiinequalities 29
1.6 The Cauchyequation 33
2 Basicnotions infunctionalanalysis 37
2.1 Linear spaces 37
2.2 Banach spaces 44
2.3 The spaceofboundedlinearoperators 63
3 Conditional expectation 80
3.1 Projectionsin Hilbertspaces 80
3.2 Definition andexistenceofconditionalexpectation 87
3.3 Properties andexamples 91
3.4 The Radon–Nikodym Theorem 101
3.5 Examples of discretemartingales 103
3.6 Convergenceofself-adjointoperators 106
3.7 ... and of martingales 112
4 Brownianmotion andHilbertspaces 121
4.1 Gaussianfamilies& the definitionofBrownianmotion 123
4.2 Complete orthonormalsequences inaHilbert space 127
4.3 Construction andbasic propertiesof Brownianmotion 133
4.4 Stochasticintegrals 139
5 Dual spaces andconvergenceofprobabilitymeasures 147
5.1 The Hahn–BanachTheorem 148
5.2 Form of linearfunctionalsinspecificBanach spaces 154
5.3 The dualofanoperator 162
5.4 Weakand weak* topologies 166
5.5 The CentralLimit Theorem 175
5.6 Weak convergencein metricspaces 178
5.7 Compactnesseverywhere 184
5.8 Notes onothermodesofconvergence 198
6 TheGelfandtransform and itsapplications 201
6.1 Banachalgebras 201
6.2 The Gelfandtransform 206
6.3 Examples of Gelfandtransform 208
6.4 Examples of explicitcalculations ofGelfandtransform 217
6.5 DensesubalgebrasofC(S) 222
6.6 Invertingthe abstractFourier transform 224
6.7 The FactorizationTheorem 231
7 Semigroupsofoperators and L′evyprocesses 234
7.1 The Banach–SteinhausTheorem 234
7.2 CalculusofBanach spacevaluedfunctions 238
7.3 Closedoperators 240
7.4 Semigroupsofoperators 246
7.5 BrownianmotionandPoissonprocesssemigroups 265
7.6 More convolutionsemigroups 270
7.7 The telegraph processsemigroup 280
7.8 Convolution semigroupsofmeasuresonsemigroups 286
8 Markov processesandsemigroupsof operators 294
8.1 Semigroupsofoperatorsrelatedto Markovprocesses 294
8.2 The Hille–Yosida Theorem 309
8.3 Generators ofstochasticprocesses 327
8.4 Approximation theorems 340
9 Appendixes 363
9.1 Bibliographicalnotes 363
9.2 Solutions and hintstoexercises 366
9.3 Some commonlyusednotations 383
References 385
Index 390
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