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Unformatted text preview: Real Analysis - Math 630 Homework Set #12 - Chapter 12 by Bobby Rohde 11-25-00 Problem 12.2 & Assume that & E i is a sequence of disjoint measurable sets and E = & E i . Then A we have that A E A E i & Proof If & E i has only one non-empty set element then this is trivial. Suppose & E i has a finite number of non-empty sets, then by the assumption of the induction hypothesis say that E & i 1 n 1 E i , conforms to the property that A E i 1 n 1 A E i . Consider E & i 1 n E i , then A E = A E E n A E E n , since E n is measurable. However E E n , and exploitning the fact that each E i is disjoint, we arrive at A E = A E n A E = i 1 n A E i . Thus the induction is proved. Hence we know for any finite sequence this property will hold. So N , A & i 1 N E i i 1 N A E i . However A & i 1 N E i A & i 1 E i , so A E A & i 1 E i i 1 N A E i . The left hand side is now independant of N so we make take the right hand side to infinity. So A E i 1 A E i , yet we know form the properties of outer measure that A E i 1 A E i . Hence A E A E i . QED. Problem 12.4 & Prove Proposition 9 - Let & be a semialgebra of sets and & a nonnegative set function defined on & with & & if & . Then & has a unique extension to a measure on the algebra generated by...
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