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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 P on (S, B(S)). For any PE P(S), we consider P"(F) := inf{P(A) : FCA, A E B(S)}, VF S S, as well as B(S)P := (Fc S: P*(F) + P*(FC) = 1}. (a) Show that P* : 2S R is an outer measure on (S, 2 S).

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 P on (S, B(S)). For any PE P(S), we consider P"(F) := inf{P(A) : FCA, A E B(S)}, VF S S, as well as B(S)P := (Fc S: P*(F) + P*(FC) = 1}. (a) Show that P* : 2S R is an outer measure on (S, 2 S).

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 P on (S, B(S)). For any PE P(S), we consider P" (F) := inf{P(A) : FSA, A E B(S)}, VF S S, as well as B(S)P := (FC S:P*(F) + P*(F©) = = 1}. (a) Show that P * : 2S → R is an outer measure on (S, 2 S ).
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 P on (S, B(S)). For any PE P(S), we consider P" (F) := inf{P(A) : FSA, A E B(S)}, VF S S, as well as B(S)P := (FC S:P*(F) + P*(F©) = = 1}. (a) Show that P * : 2S → R is an outer measure on (S, 2 S ).
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