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Let S be a topological space, B(S) be the Borel o-algebra on S, and P(S) be the set of all probability measures on (S, B(S)). For any E P(S), we consider P*(F) := inf{(A) : F≤A, A E B(S)}, VF S S, as well as B(S) P := {FCS : P*(F) + P*(Fc) = 1}. solution to c please (a) Show that P* : 2S → R is an outer measure on (S, 2S) (b) Show that B(S) is exactly the collection of all λ-sets under the outer measure P* . (Note: Recall the definition of a λ-set on p. 5 of Ch I of lecture notes). (c) Show that B(S) is a o-algebra and it coincides with B(S)* := {F C S: BE, G E B(S) s.t. ECFCG and (G\ E) = 0}.

Let S be a topological space, B(S) be the Borel o-algebra on S, and P(S) be the set of all probability measures on (S, B(S)). For any E P(S), we consider P*(F) := inf{(A) : F≤A, A E B(S)}, VF S S, as well as B(S) P := {FCS : P*(F) + P*(Fc) = 1}. solution to c please (a) Show that P* : 2S → R is an outer measure on (S, 2S) (b) Show that B(S) is exactly the collection of all λ-sets under the outer measure P* . (Note: Recall the definition of a λ-set on p. 5 of Ch I of lecture notes). (c) Show that B(S) is a o-algebra and it coincides with B(S)* := {F C S: BE, G E B(S) s.t. ECFCG and (G\ E) = 0}.

Advanced Engineering Mathematics
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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Let S be a topological space, B(S) be the Borel o-algebra on S, and P(S) be the set of all probability measures on (S, B(S)). For any E P(S), we consider p*(F) := inf{(A) : FCA, A E B(S)}, VF S S, as well as B(S) P := {F C S: P*(F) + P*(Fc) = 1}. %3D solution to c please (a) Show that P* : 2S → Ris an outer measure on (S, 2S) (b) Show that B(S) is exactly the collection of all X-sets under the outer measure P* . (Note: Recall the definition of a A-set on p. 5 of Ch I of lecture notes). (c) Show that B(S) is a o-algebra and it coincides with B(S)* := {F CS:3E, G E B(S) s.t. EC FSG and (G \ E) = 0}. %3D
Transcribed Image Text:Let S be a topological space, B(S) be the Borel o-algebra on S, and P(S) be the set of all probability measures on (S, B(S)). For any E P(S), we consider p*(F) := inf{(A) : FCA, A E B(S)}, VF S S, as well as B(S) P := {F C S: P*(F) + P*(Fc) = 1}. %3D solution to c please (a) Show that P* : 2S → Ris an outer measure on (S, 2S) (b) Show that B(S) is exactly the collection of all X-sets under the outer measure P* . (Note: Recall the definition of a A-set on p. 5 of Ch I of lecture notes). (c) Show that B(S) is a o-algebra and it coincides with B(S)* := {F CS:3E, G E B(S) s.t. EC FSG and (G \ E) = 0}. %3D
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