STAT
A Probability Path.pdf

# A identify the complex plane with jr 2 show bc t31r 2

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(a) Identify the complex plane with JR 2 Show B(C) = T3(1R 2 ) n c.

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3.4 Exercises 87 (b) Show that B(C) is generated by arcs of C. (c) Show Jl is invariant with respect to rotations. This means, if Seo C C via then Jl = Jl o Si;/. (d) If you did not define Jl as the induced image of Lebesgue measure on the unit interval, how could you define it by means of the extension theorems? 11. Let (Q, B, P) be ([0, 1], 8([0, 1]), >..) where >.. is Lebesgue measure on [0, 1] . Define the process {X 1, 0 t 1} by X 1 (w) = 1 0, 1, if t =ft w, if t = w. Show that each X 1 is a random variable. What is the a-field generated by {X 1, 0 t 1}? 12. Show that a monotone real function is measurable. 13. (a) If X is a random variable, then a(X) is a countably generated a-field. (b) Conversely, if B is any countably generated a -field, show B = a(X) for some random variable X. 14 . A real function f on the line is upper semi-continuous (usc) at x, if, for each E, there is a 8 such that lx - yl < 8 implies that f(y) < f(x) +E. Check that if f is everywhere usc, then it is measurable. (Hint: What kind of set is {x : f (x) < t} ?) 15 . Suppose -00 < a b < oo . Show that the indicator function l(a,bJ(x) can be approximated by bounded and continuous functions; that is, show that there exist a sequence of continuous functions 0 fn 1 such that fn 1(a, b) pointwise. Hint: Approximate the rectangle of height 1 and base (a, b] by a trapezoid of height 1 with base (a, b + n- 1 ] whose top line extends from a+ n- 1 to b. 16 . Suppose B is a a-field of subsets of!R. Show B(IR) c B iff every real valued continuous function is measurable with respect to B and therefore B(IR) is the smallest a-field with respect to which all the continuous functions are measurable.
88 3. Random Variables, Elements, and Measurable Maps 17. Functions are often defined in pieces (for example, let f (x) be x 3 or x- 1 as x 0 or x < 0), and the following shows that the function is measurable if the pieces are. Consider measurable spaces (Q, B) and (Q', B') and a map T : Q t--+> Q'. Let At. Az, ... be a countable covering of Q by B sets. Consider the a- field Bn = {A : A C An, A E B} in An and the restriction Tn ofT to An. Show that Tis measurable B!B' iff Tn is measurable Bn!B' for each n. 18. Coupling. If X andY are random variables on (Q, B), show sup IP[X e A]- P[Y e A]l P[X # Y] . AEB 19. Suppose T : (Qt. Bt) t--+> (Qz, Bz) is a measurable mapping and X is a random variable on !'21 . Show X e a(T) iff there is a random variable Y on (Qz, Bz) such that X(wt) = Y(T(wi)), 'v'w1 E !:'21 . 20. Suppose {X, , t 0} is a continuous time stochastic process on the proba- bility space (Q, B, P) whose paths are continuous. We can understand this to mean that for each t 0, X, : Q t--+> lR is a random variable and, for each w e Q, the function t t--+> X 1 (w) is continuous; that is a member of C[O, oo). Let r : Q t--+> [0, oo) be a random variable and define the process stopped at r as the function X, : Q t--+> [0, oo) defined by X,(w) := Xr(w)(w), wE Q.

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