8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n )) 8 arbitrary set. K is Cousider...
Let C,C Є F where F is a sigma algebra on Ω with a probability measure P. Show that F1={ⱷ, Ω ,C,Cc} and F2={ ⱷ, Ω ,D,Dc } are independent iff C and D are independent?
(1) Let (2, A, i) be a measure space {AnE A E A} is a (a) Fix E E A. Prove that Ap 0-algebra of E, contained in A. (b) Let /i be the restriction of /u to Ap. Prove that ip is a measure on Ap. (c) Suppose that f : O -» R* is measurable (with respect to A). Let g the restriction of f to E. Prove that g : E -> R* is measurable (with respect...
Let L be a regular language on sigma = {a, b, d, e}. Let L' be the set of strings in L that contain the substring aab. Show that L' is a regular language.
(1) Let (, A, /i) be a measure space = {AnE: A E A} is a o-algebra of E, contained in (a) Fix E E A. Prove that AE A. (b) Let be the restriction of u to AE. Prove that uE is a measure on Ag (c) Suppose that f -> R* is measurable (with respect to A). Let g = f\e be the restriction of f to E. Prove that g E ->R* is measurable (with respect to...
(1) Let (, A, i) be a measure space. {AnE: Ae A} is a o-algebra of E, contained in (a) Fix E E A. Prove that Ap = A. (b) Let uE be the restriction of u to AĘ. Prove that iE is a measure on Ag. (c) Suppose that f : Q -» R* is measurable (with respect to A). Let g = the restriction of f to E. Prove that g : E ->R* is measurable (with respect...
1. Let the random variables X.X be dependent and denote the sigma algebra as g" o(x,, ,X"} Assume that E(X)-μ and E(x, II,-1)-μ, 1-2, .., n, where μ is a constant. Please find ETTx
Let 2 N (1,2,3,...} be a sample space and F-2N a sigma algebra. . . . . } with F = 2Ω. Define P a. Consider the sample space Ω-{1, 2, 3 on (2, F) as follows: Show that (2,F, P) is a probability space. b. Find the values of B for which the following P defined on (2, F) is a probability measures: k2k
Problem 1.11. Let P be a probability measure on R, equipped with the Borel ơ-algebra. Let F(x)-P((-00,2]). P rove that f is non-decreasing right-continuous, F(x) → 0 as x →-00, and F(x) → 1 as x → oo. Prove that if P and Q are two probability measures such that P((-oo, x Q((-00,x]) for all x rational, then P , ie. P(A) = Q(A) for any Borel- measurable set A.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular. (3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.