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Unformatted text preview: 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 ( n ) and the measure of the lim inf of the sets 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 Ill discuss the main elements, in a streamlined way. Ill be perfectly satisfied if you master the topics discussed in lecture. One example is the concept of a measurable mapping. Well simply discuss measurable R-valued functions, that is, measurable mappings from some measure space to the extended real numbers R = [- , + ], and Ill give a more direct definition. For present purposes we wont 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 spaces (which are a special class of topological spaces); for our purposes metric spaces are sufficiently general.purposes metric spaces are sufficiently general....
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This note was uploaded on 05/07/2008 for the course MATH 105 taught by Professor Michaelchrist during the Spring '04 term at University of California, Berkeley.
- Spring '04
- Real Numbers