Measure theory/ 4, Topological measure spaces by D.H. Fremlin

By D.H. Fremlin

Fremlin D.H. degree conception, vol.4 (2003)(ISBN 0953812944)(945s)-o

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Extra resources for Measure theory/ 4, Topological measure spaces

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When we come to look more closely at products of Radon probability spaces we shall need to consider this point again (417Q, 417Xq). In fact some of the ideas of 412U-412V are not restricted to the product measures considered there. Other measures on the product space will have inner regularity properties if their images on the factors, their ‘marginals’ in the language of probability theory, are inner regular; see 412Xn. I will return to this in §454. This section is almost exclusively concerned with inner regularity.

There is one special difficulty in 412V: in order to ensure that there are enough compact measurable sets in X = i∈I Xi , we need to know that all but countably many of the Xi are actually compact. When we come to look more closely at products of Radon probability spaces we shall need to consider this point again (417Q, 417Xq). In fact some of the ideas of 412U-412V are not restricted to the product measures considered there. Other measures on the product space will have inner regularity properties if their images on the factors, their ‘marginals’ in the language of probability theory, are inner regular; see 412Xn.

We know that χKn ≤ Kn ⊆ i≤n n i=0 χLi , so Li ⊆ Ln ⊆ Kn+1 ; accordingly we have K0 ⊆ K1 ⊆ . . ⊆ Kn ⊆ Kn+1 . Next, n+1 n χKi = i=0 n−1 χLi + χLn+1 = i=0 χLi + χ(Ln ∩ Ln+1 ) + χ(Ln ∪ Ln+1 ) i=0 n−1 = n χLi + χLn + χLn+1 = i=0 n+1 i=0 n λKi ≥ i=0 n+1 χLi + χKn+1 = χKi , i=0 n−1 λLi + λLn+1 = i=0 λLi + λ(Ln ∩ Ln+1 ) + λ(Ln ∪ Ln+1 ) i=0 n−1 ≥ λLi + λLn + λLn+1 i=0 (using the hypothesis on λ) n = n+1 λLi + λKn+1 ≥ i=0 λKi , i=0 Q so we have an appropriate family K0 , . . , Kn+1 , and the induction continues.

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