Foundations of the theory of probability

§ ProbabilitiesinInfinite-dimensionalSpaces 29 tionfunction ^^...^(fli,a 2 , ... , a w). Itisobviousthatfor everyBorelcylinder ...

30 III. RandomVariables Proof. Giventhedistributionfunctions ^ 1/ u t ... / . B , satisfying thegeneralconditionsofChap.II, § 3, ...

§ ProbabilitiesinInfinite-dimensionalSpaces 31 Letus nowprovethatthefieldofprobability (JP, P) satisfies allthe AxiomsI VI.Ax ...

32 III.RandomVariables occur,i.e. ^ = ^,. ..,.»(£»)• Forbrevityweset ^, t...Mn(B) = P n (B); then, obviously P n (B n) =?(A n ) ...

§ 5.EquivalentRandomVariables;VariousKindsof Convergence 33 Asthelimitofthesequence (x^, 4 Wl) ,•.•, #i Wi) ), i = 1,2,3, .. . , ...

34 III. Random Variables therandomvariablexisuniquelydennedexceptforequivalence. Todeterminesucharandomvariablewe set limx n n o ...

§ 5.Equivalent RandomVariables;VariousKindsofConvergence 35 alsoconvergestox inprobability.LetAbetheconvergenceset ofthesequence ...

36 III.RandomVariables SinceF(a') andF(a") convergetoF(a) fora' — *aanda" — a, itfollowsfrom (3) and (4) that limF B (a) = F(a), ...

ChapterIV MATHEMATICAL EXPECTATIONS 1 § Abstract LebesgueIntegrals Let#bearandomvariableandAasetofgf.Letusform,fora positiveA, ...

38 IV. MathematicalExpectations morethanacountable numberofnon-intersecting sets A n ofgf, then r _ , JxPXdE)=£jxP(dE). A nAn II ...

§ AbsoluteandConditionalMathematicalExpectations 39 IX. If (3) holds for every set A of gf, then x and y are equivalent. Fromt ...

40 IV. MathematicalExpectations Thesecondlineisnothingmorethantheusualdefinitionofthe Stieltjesintegral +«> jadFW(a)= E(*). ( ...

§ 2.AbsoluteandConditionalMathematicalExpectations 41 E{x) = ft. .. fx(ult u 2 ,..., u n ) P<«i.«*.-.«> («*#») . (4) We ha ...

42 IV.MathematicalExpectations § TheTchebycheffInequality Let f(x) bea non-negativefunction ofa realargumentx, whichforx ^ ane ...

§4. SomeCriteriaforConvergence 43 E (/(*))==//(*) P{dE) =jf(x) P(dE) +//(*) P(dE) ^f{a)P(\x\< a) + KP()x\ > a) £/(«)+ KP(| ...

44 IV.MathematicalExpectations FromIIandIV,weobtaininparticular V. Inorderthat sequence (1) converge inprobabilitytox, itissuffi ...

§ 5.Differentiationand Integration of MathematicalExpectations 45 matical expectation E[x'(t)] exists (Property VII of mathe- ma ...

46 IV. MathematicalExpectations Therefore,S*convergestoE(J),fromwhichresultstheequation b Ex(t)dt = limS* n = E(/). /' TheoremII ...

ChapterV CONDITIONAL PROBABILITIES AND MATHEMATICAL EXPECTATIONS § ConditionalProbabilities In § 6,ChapterI,wedennedtheconditi ...

48 V. ConditionalProbabilitiesand MathematicalExpectations PM(A) > 0.The function P U (B) ofuthusdeterminedtowithin equivalen ...