III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T

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1 III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). Theorem 1.3 (Lebesgue ) lim n f n = f µ-a.e. g L 1 (µ) such that n N, x M, f n (x) g(x) ( ) (1.1), lim n f n = f in L 1 (µ). Proposition 1.4 µ,. Example 1.5 Example 1.6 µ Proposition 1.4.,. Proposition 1.4.,. Proposition 1.7. Proposition 1.8 (Borel Cantelli ) n=1 ( µ(a n ) < µ n=1 k=n ) A k = 0. Proposition 1.9,. Lemma 1.10 {x n } x, {x n } {x nk } x {x nkl }. Proposition 1.11 (Lebesgue ) Theorem 1.3, µ-a.e. in µ OK. Proposition 1.12 µ, lim n f n = f in µ (1.1) lim sup f n (x) µ(dx) = 0 (1.2) a n N {x M f n(x) >a}, lim n f n = f in L 1 (µ). 1

2 Definition 1.13 µ, L 0 (M, F, µ): d(f, g) = φ( f(x) g(x) )µ(dx) M (φ R 0, 0, φ(0) = 0, φ(x)/x R 0 ). Proposition 1.14 (1) d. (2), d. Remark ,,. A, A ( ) σ[a] Definition 2.1 M P (i) M P (ii) A, B P = A B P, P., D (I) M D (II) A, B D, A B = B \ A D (III) A n D (n N), A n A ( A n A n+1 A = n=1 A n) = A D, D Dynkin. A Dynkin δ[a]. Lemma 2.2 M B, Dynkin. Theorem 2.3 (Dynkin ) Corollary 2.4 P, P Dynkin σ[p]. δ[p] = σ[p]. Theorem 2.5 (Carathéodory ) A 2 M (M ) ν(a) [0, ] (i) ν( ) = 0 (ii) A 1 A 2 = ν(a 1 ) ν(a 2 ) (iii) ν( k=1 A k) k=1 ν(a k), ν B, ν B., B ν ν(b A) + ν(b c A) = ν(a), A M. Theorem 2.6 (Hopf ) M A µ. ( ) B = σ[a] µ ( ) k=1 A k = A A 1 A 2 lim k µ(a k ) = 0., B d(b 1, B 2 ) = µ(b 1 B 2 ) A, µ B. 2

3 3 Hahn, Jordan Definition 3.1 (M, F) (R-, ) Φ : F R Theorem 3.2 ( Hahn ) (M, F) Φ, Φ(A P ) 0, A F (P Φ ) Φ(A P c ) 0, A F (P c Φ ). P F. M = P P c Φ M Hahn. Φ (A) = sup { i Φ(A i) A = i A i ( ) }, A F Lemma 3.3 Φ Hahn M = P P c (1) Φ (A) = Φ(A P ) Φ(A P c ), A F. (2) Φ, Φ(A) Φ (A), A F. Φ Φ. Proposition 3.4 (Hahn ) Φ M Hahn M = P 1 P c 1 = P 2 P c 2 = Φ (P 1 P 2 ) = Φ (P c 1 P c 2 ) = 0., : A B = (A B c ) (A c B). Proposition 3.5 : Φ(P ) = sup{φ(a) A F}, Φ(P c ) = inf{φ(a) A F}. Definition 3.6 ( Jordan ) Φ Hahn P P c Φ + (B) = Φ(B P ), Φ (B) = Φ(B P c ), B F, Φ +, Φ, Φ = Φ + Φ, Φ = Φ + + Φ. Proposition 3.7 (Jordan ) Φ Hahn M = P 1 P1 c = P 2 P2 c Φ = Φ + 1 Φ 1 = Φ+ 2 Φ 2, Φ+ 1 = Φ+ 2, Φ 1 = Φ 2. 3

4 4 Radon Nikodym Definition 4.1 (ν µ), (ν µ), (ν µ). Φ µ (Φ µ). Lemma 4.2 µ ν ν = ν 1 + ν 2 where ν 1 µ, ν 2 µ (4.1). Lemma 4.3 ν, ν ν. Proposition 4.4 ν, ν (4.1). Lemma 4.5 µ f ν(b) = f(x)µ(dx), B F B, ν ν µ ( ν(dx) = f(x)µ(dx) ν = fµ ). : (gf)µ = g(fµ). Theorem 4.6 (Radon Nikodym ) µ, ν ν µ, ν(b) = f(x)µ(dx), B F B f µ-a.e.. f Radon Nikodym ( ). Lemma 4.7 Φ Jordan Φ = Φ + Φ, Φ µ Φ + µ Φ µ. Theorem 4.8 Φ µ Φ µ Radon Nikodym, Radon Nikodym L 1 (µ; R). Example 4.9 Radon Nikodym., 0. Remark Remark 4.11 Radon Nikodym L p (1 p < ) Lebesgue 4

5 5 Riesz Markov Definition 5.1 E Borel B(E), (E, B(E)) E Borel. C(E) = C(E; C) : E C-, C(E; R) : E R-. C c (E) = {f C(E) suppf }, C c (E; R) = {f C(E; R) suppf } (E f : suppf = {x E f(x) 0}). C(E)( C(E; R)) ψ : ψ(αf + βg) = αψ(f) + βψ(g), α, β C ( R), : f 0 = ψ(f) 0. Definition 5.2 E µ, O, µ(o) = sup{µ(k) K O, K } Borel B, µ(b) = inf{µ(o) O B, O }. Borel Radon. Lemma 5.3 K Borel, Radon. Lemma 5.4 X, C O. (1) C G G O G G. (2) X f, 0 f 1, C f = 1, suppf O. Remark 5.5 X,. Theorem 5.6 (Riesz Markov ) K. C(K; R) ψ K Borel µ, : ψ(f) = f(x)µ(dx), f C(K; R). (5.1) K Remark 5.7,. Theorem 5.8 (Riesz Markov ) C c (R; R) ψ R Radon µ : ψ(f) = f(x)µ(dx), f C c (R; R). (5.2) R Remark 5.9 C-, ψ R- µ. f = Ref + iimf, Ref Imf (5.1) (5.2), f. 5

6 [1] R Borel. (1) M {f α } α A, f α M (, )., A. (2) R n R k f k : f k (x 1,, x n ) = x k., f k R n R n Borel. [2] M Borel.. [3],,,.. ( ). ( ). 6

7 PART II 6 Definition 6.1 (Ω, F, P ) X : Ω S, (S, S). X : Ω S P X P = P X X. X : Ω R F (x) = P X ((, x]) = P (X x) X : Ω R P X Lebesgue, Radon Nikodym f X X. X, Y : Ω S (X, Y ) P = P (X,Y ) (S S ). X : Ω R E[X]: E[X] = X(ω)P (dω) = xp X (dx). Ω R, X (E[ X ] < ). Theorem 6.2 (, ) X : Ω S, f : S R 0 E[f(X)] = f(x)p X (dx). Remark 6.3, f C-. S Proposition 6.4 Z E[Z] = 0 P (Z > t)dt. 7, Prokhorov Definition 7.1,,,, Lemma 7.2 R, R., (S, d), (S, d) (S, d) d. Lemma 7.3 K, Banach C(K). Lemma 7.4 (1 ) K K = U 1 U n, ϕ 1,, ϕ n C(K). (i) 0 ϕ k (x) 1, suppϕ k U k (k = 1,..., n), (ii) ϕ 1 (x) + + ϕ n (x) = 1 (x K). 7

8 Example 7.5 l sup-. Definition 7.6 S µ n µ (n ): f(x)µ n (dx) = f(x)µ(dx), f C b (S). lim n S S Theorem 7.7 K, (K, B(K)) P(K). Theorem 7.8 ( ) R, C b (R), C u (R). µ n, µ P(R). (i) lim n R f(x)µ n(dx) = R f(x)µ(dx), f C b(r). (ii) lim n R f(x)µ n(dx) = R f(x)µ(dx), f C b(r) C u (R). (iii) lim sup n µ n (F ) µ(f ), F : R. (iv) lim inf n µ n (O) µ(o), O : R. (v) lim n µ n (B) = µ(b), B B(R) : µ( B) = 0. (vi) lim n F n (x) = F (x), x R : F., F n, F µ n, µ. Remark 7.9 Theorem 7.8, (ii) R ( ). (ii), OK. Definition 7.10 Corollary 7.11,,,., =. Example 7.12 {x n } x, δ xn δ x., Theorem 7.8 (iii), (iv). Definition 7.13 S Ŝ,. Remark 7.14 P(R). R d ( R, d). Lemma 7.3 C u (R, d) = C( R, d) {f n } n N, τ(µ, ν) = n=1 1 ( 2 n R f n (x)µ(dx) R ) f n (x)ν(dx) 1, µ, ν P(R), Theorem 7.8, τ. Theorem 7.15 (Prokhorov ) (C) Q P(R). P(R) Q. (T) Q (tight), ɛ > 0, C : R, µ Q, µ(c) 1 ɛ. Remark Brown, R C([0, ); R) Prokhorov 8

9 8 Definition 8.1 ( ), (Fubini ) Definition 8.2,, Theorem 8.3 (i) X, Y X, Y. (ii) E[f(X)g(Y )] = E[f(X)]E[g(Y )] ( f, g C b (R)) (iii) E[e i(sx+ty ) ] = E[e isx ]E[e ity ] ( s, t R) (iv) P (X,Y ) = P X P Y Theorem 8.4 (Ω n, F n ) µ n µ = n=1 µ n (Ω, F) = ( n=1 Ω n, n=1 F n). Remark 8.5, : (Ω, F, P ), (S, S), S- {X n } n N, F G = σ[x n n N], (S, S ) µ, n X n : S S. Theorem 8.6 (Borel Cantelli ( 2) ) {A n } n N, ( ) P (A n ) = = P A k = P ( A k i.o. ) = 1. Definition 8.7 n=1 n=1 k=n {X n } n N σ[x 1, X 2, ] (tail event) : T = n=1 σ[x n, X n+1, ] = X 1,, X n. (exchangeable event) : E = X 1,, X n. Remark 8.8. T E. E \ T, X X n Theorem 8.9 (Kolmogorov 0-1 ) {X n } n N, A T = P (A) = 0 or 1. Theorem 8.10 (Hewitt Savage 0-1 ) 1. {X n } n N, A E = P (A) = 0 or Theorem 8.11 ( (dichotomy)) (Ω n, F n ) µ n ν n µ = n=1 µ n, ν = n=1 ν n, µ ν µ ν, ( dν ) n µ ν 1 dµ n <. dµ n n=1 Ω n 9

10 9, {X n } n N (Ω, F, P ), S n = X 1 + X X n. Proposition S n a.s. a.s.., S n Theorem 9.2 (Kolmogorov ) E[X n ] = 0, V (X n ) <, P ( max k=1,...n S k a ) 1 a 2 V (S n), n N, a > 0. Theorem 9.3 (Kolmogorov : strong Law of Large Numbers) (I) {X n } E[ X 1 ] <, (II) n=1 n 2 V (X n ) <, S n E[S n ] lim = 0 a.s. n n Remark 9.4 {X n } E[X 4 1 ] <, Borel Cantelli. Remark 9.5 {X n } E[X 2 1 ] <, (weak), Chebyshev. Theorem 9.6 ( : Central Limit Theorem) {X n } m = E[X 1 ], v = V (X 1 ), S n E[S n ] lim = Z ( ). n n, Z 0 v N(0, v). 10

11 [1] M Hausdorff Borel.. [2] (Corollary7.11). [3] X, Y ( ), ( ). ( ) E[f(X)g(Y )] = E[f(X)]E[g(Y )] ( f, g C b (R)) ( ) P (X,Y ) = P X P Y [4], A, B, C. ( ) A, B, C, A, B, C. [5],,,.. ( ) ( ). 11

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 ( . 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1