Let (Ω,F,P) be a probability space. (a) Show that the set Fa.s. := {A ∈ F; P(A) = 0 or 1} is a σ-algebra. (b) Let X : (Ω, Fa.s.) → (R, B) be a random variable. Prove that there is a unique x ∈ R such that P({X = x}) = 1.

Let (Ω,F,P) be a probability space. (a) Show that the set Fa.s. := {A ∈ F; P(A) = 0 or 1} is a σ-algebra. (b) Let X : (Ω, Fa.s.) → (R, B) be a random variable. Prove that there is a unique x ∈ R such that P({X = x}) = 1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let (Ω,F,P) be a probability space.

(a) Show that the set Fa.s. := {A ∈ F; P(A) = 0 or 1} is a σ-algebra.

(b) Let X : (Ω, Fa.s.) → (R, B) be a random variable. Prove that there is a unique x ∈ R such that P({X = x}) = 1.

Hint: Be careful, there are plenty of real valued random variables such thatP({X=x})=0forallx∈R. TryusetheCDFofX.

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