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Unformatted text preview: Real Analysis  Math 630 Homework Set #12  Chapter 12 by Bobby Rohde 112500 Problem 12.2 & Assume that & E i Â¡ is a sequence of disjoint measurable sets and E = & E i . Then Â¢ A we have that Â£ Â¤ Â¡ A Â¢ E Â£ Â¥ Â¤ Â£ Â¤ Â¡ A Â¢ E i Â£ & Proof If & E i Â¡ has only one nonempty set element then this is trivial. Suppose & E i Â¡ has a finite number of nonempty sets, then by the assumption of the induction hypothesis say that E Â¢ Â£ & i Â£ 1 n Â¤ 1 E i , conforms to the property that Â¥ Â¦ Â¡ A Â¢ E Â¢ Â£ Â£ Â¤ i Â£ 1 n Â¤ 1 Â¥ Â¦ Â¥ A Â¢ E i Â¦ . Consider E Â£ & i Â£ 1 n E i , then Â¥ Â¦ Â¥ A Â¢ E Â¦ = Â¥ Â¦ Â¥ A Â¢ E Â¢ E n Â¦ Â§ Â¥ Â¦ Â¥ A Â¢ E Â¢ E n Â¨ Â¦ , since E n is measurable. However E Â© E n , and exploitning the fact that each E i is disjoint, we arrive at Â¥ Â¦ Â¥ A Â¢ E Â¦ = Â¥ Â¦ Â¥ A Â¢ E n Â¦ Â§ Â¥ Â¦ Â¡ A Â¢ E Â¢ Â£ = Â¤ i Â£ 1 n Â¥ Â¦ Â¥ A Â¢ E i Â¦ . Thus the induction is proved. Hence we know for any finite sequence this property will hold. So Âª N , Â¥ Â¦ Â¥ A Â¢Â¥& i Â£ 1 N E i Â¦Â¦ Â£ Â¤ i Â£ 1 N Â¥ Â¦ Â¥ A Â¢ E i Â¦ . However A Â¢Â¥& i Â£ 1 N E i Â¦ Â« A Â¢ Â¥& i Â£ 1 Â¬ E i Â¦ , so Â¥ Â¦ Â¥ A Â¢ E Â¦ Â£ Â¥ Â¦ Â¥ A Â¢ Â¥& i Â£ 1 Â¬ E i Â¦Â¦ Â Â¤ i Â£ 1 N Â¥ Â¦ Â¥ A Â¢ E i Â¦ . The left hand side is now independant of N so we make take the right hand side to infinity. So Â¥ Â¦ Â¥ A Â¢ E Â¦ Â Â¤ i Â£ 1 Â¬ Â¥ Â¦ Â¥ A Â¢ E i Â¦ , yet we know form the properties of outer measure that Â¥ Â¦ Â¥ A Â¢ E Â¦ Â® Â¤ i Â£ 1 Â¬ Â¥ Â¦ Â¥ A Â¢ E i Â¦ . Hence Â¥ Â¦ Â¥ A Â¢ E Â¦ Â£ Â¤ Â¥ Â¦ Â¥ A Â¢ E i Â¦ . QED. Problem 12.4 & Prove Proposition 9  Let & be a semialgebra of sets and & a nonnegative set function defined on & with &Â¡ Â¢ & if Â¡ Â£ & Â¡ . Then & has a unique extension to a measure on the algebra Â¡ generated by...
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 Spring '08
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 Math, Sets, Empty set, Natural number, Finite set, Basic concepts in set theory, Ei Bi C

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