This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Real Analysis  Math 630 Homework Set #12  Chapter 12 by Bobby Rohde 112500 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 nonempty set element then this is trivial. Suppose & E i has a finite number of nonempty 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...
View
Full
Document
 Spring '08
 Staff
 Math, Sets

Click to edit the document details