P is regular if all Borel sets are regular. Define C to be the collection of regular sets. (a) Show R e C, Ø e C. (b) Show C is closed under complements and countable unions. (c) Let F(R) be the closed subsets of Rk. Show

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27. Regular measures. Consider the probability space (R, B(Rk), P). A Borel
set A is regular if
P(A) = inf(P(G) : G Ɔ A, G open,}
and
P(A) = sup(P(F) : FCA, F closed.)
P is regular if all Borel sets are regular. Define C to be the collection of
regular sets.
(a) Show Rk E C, ØE C.
(b) Show C is closed under complements and countable unions.
(c) Let F(R) be the closed subsets of Rk. Show
F(R*) CC.
(d) Show B(R) C C; that is, show regularity.
(e) For any Borel set A
P(A) = sup{P(K): K CA, K compact.)
Transcribed Image Text:27. Regular measures. Consider the probability space (R, B(Rk), P). A Borel set A is regular if P(A) = inf(P(G) : G Ɔ A, G open,} and P(A) = sup(P(F) : FCA, F closed.) P is regular if all Borel sets are regular. Define C to be the collection of regular sets. (a) Show Rk E C, ØE C. (b) Show C is closed under complements and countable unions. (c) Let F(R) be the closed subsets of Rk. Show F(R*) CC. (d) Show B(R) C C; that is, show regularity. (e) For any Borel set A P(A) = sup{P(K): K CA, K compact.)
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