[...]... completions S µ and T ν 9 Fix a measure space (S, S, µ) and a - eld T ⊂ S, let S µ denote the µ-completion of S, and let T µ be the - eld generated by T and the µ-null sets of S µ Show that A ∈ T µ iff there exist some B ∈ T and N ∈ S µ with A∆B ⊂ N and µN = 0 Also, show by an example that T µ may be strictly greater than the µ-completion of T 10 State Fubini’s theorem for the case where µ is any - nite... functions and Dirichlet’s problem Green functions as occupation densities sweeping and equilibrium problems dependence on conductor and domain time reversal capacities and random sets 22 Predictability, Compensation, and Excessive Functions accessible and predictable times natural and predictable processes Doob–Meyer decomposition quasi–left-continuity compensation of random measures excessive and superharmonic... 455 Historical and Bibliographical Notes 464 Bibliography 486 Indices Authors Terms and Topics Symbols 509 Chapter 1 Elements of Measure Theory - elds and monotone classes; measurable functions; measures and integration; monotone and dominated convergence; transformation of integrals; product measures and Fubini’s theorem; Lp spaces and projection; measure spaces and kernels Modern probability theory... of Brownian motion 20 One-Dimensional SDEs and Diffusions 371 weak existence and uniqueness pathwise uniqueness and comparison scale function and speed measure time-change representation boundary classification entrance boundaries and Feller properties ratio ergodic theorem recurrence and ergodicity 21 PDE-Connections and Potential Theory 390 backward equation and Feynman–Kac formula uniqueness for SDEs... introduced Random elements are of interest in a wide variety of spaces A random element in S is called a random variable when S = R, a random vector when S = Rd , a random sequence when S = R∞ , a random or stochastic process when S is a function space, and a random measure or set when S is a class of measures or sets, respectively A metric or topological space S will be endowed with its Borel - eld B(S)... spaces S and T is measurable with respect to the Borel - elds B(S) and B(T ) Proof: Use Lemma 1.4, with C equal to the topology in T ✷ Here we insert a result about subspace topologies and - elds, which will be needed in Chapter 14 Given a class C of subsets of S and a set A ⊂ S, we define A ∩ C = {A ∩ C; C ∈ C} Lemma 1.6 (subspaces) Fix a metric space (S, ρ) with topology T and Borel - eld S, and let... is said to be degenerate, and we note that µ = cδs for some s ∈ S and c ≥ 0 More generally, a measure µ is said to have an atom at s ∈ S if {s} ∈ S and µ{s} > 0 For any locally finite measure µ on some σ-compact metric space S, the set A = {s ∈ S; µ{s} > 0} is clearly measurable, and we may define the atomic and diffuse components µa and µd of µ as the restrictions of µ to A and its complement We further... The following lemma justifies the formula and provides some further useful information Lemma 1.38 (kernels and functions) Fix three measurable spaces (S, S), (T, T ), and (U, U) Let µ and ν be probability kernels from S to T and from S × T to U , respectively, and consider two measurable functions f : S × T → R+ and g : S × T → U Then 20 Foundations of Modern Probability (i) µs f (s, ·) is a measurable... decomposition 409 xii Foundations of Modern Probability 23 Semimartingales and General Stochastic Integration 433 2 predictable covariation and L -integral semimartingale integral and covariation general substitution rule Dol´ans’ exponential and change of measure e norm and exponential inequalities martingale integral decomposition of semimartingales quasi-martingales and stochastic integrators Appendices... elements and processes; distributions and expectation; independence; zero–one laws; Borel–Cantelli lemma; Bernoulli sequences and existence; moments and continuity of paths Armed with the basic notions and results of measure theory from the previous chapter, we may now embark on our study of probability theory itself The dual purpose of this chapter is to introduce the basic terminology and notation and . existence moments and continuity of paths 3. Random Sequences, Series, and Averages 39 convergence in probability and in L p uniform integrability and tightness convergence in distribution convergence of random. Poisson and Pure Jump-Type Markov Processes 176 existence and characterizations of Poisson processes Cox processes, randomization and thinning one-dimensional uniqueness criteria Markov transition and. integration; monotone and dominated convergence; transfor- mation of integrals; product measures and Fubini’s theorem; L p - spaces and projection; measure spaces and kernels Modern probability theory