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hw7 - Mathematics 105 Spring 2004 Problem Set VII Remark on...

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Mathematics 105, Spring 2004 — Problem Set VII Remark on problem 3.1.11: If ( x n ) is a sequence of real numbers, lim n →∞ x n is a common alternative notation for lim sup n →∞ x n ; likewise lim is denotes the lim inf. In this problem analogous notions of lim sup and lim inf are defined for sequences of sets. Inequality (i) deals with a relation between the lim inf of the real numbers μ n ) and the μ measure of the lim inf of the sets Γ n ; thus the first symbol lim refers to the notion for sets, while the second instance of the same symbol refers instead to the notion for real numbers. A couple of other miscellaneous remarks on the text: (i) The discussion in §§ 3.1 and 3.2 is a bit more abstract and general than is really necessary for us. In class I’ll discuss the main elements, in a streamlined way. I’ll be perfectly satisfied if you master the topics discussed in lecture. One example is the concept of a measurable mapping. We’ll simply discuss measurable R -valued functions, that is, measurable mappings from some measure space to the extended real numbers R = [ -∞ , + ], and I’ll give a more direct definition. For present purposes we won’t need to know about tensor products of measure spaces or of measurable mappings (this will come up later, when we discuss the connection between Lebesgue measure/integration on R n , R m , and R m + n . (ii) Wherever Stroock mentions topological spaces, you should assume that he is talking about metric
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