page5 - Math 172 Homework (Spring 2009) 5 Assignment 5:...

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Unformatted text preview: Math 172 Homework (Spring 2009) 5 Assignment 5: Assigned Tue 04/28. Due Tue 05/05. (Will be accepted Thu 05/07.) Browse through Jones Chapter 11, and do whatever problems you find interesting. 1. Chapter 6: 21, 25, 28, 29, 30, 31. 2. Let X negationslash = be a set, M a -algebra on X and let , be two positive measures on M . (a) Let A 2 X be an algebra such that ( A ) = ( A ) for all A A . If ( X ) = ( X ) < , then show that ( A ) = ( A ) for all A ( A ). (b) Show that the previous subpart need not be true if ( X ) = ( X ) = . (c) Show that the conclusion of part (a) holds if X = A i , with A i A , and ( A i ) = ( A i ) < . (d) Let M be an n n matrix and A L ( R n ), show that ( MA ) = | det M | ( A ). [ Hint: First assume M is invertible. Define ( A ) = | det M | ( M- 1 A ), and verify is a measure. From standard linear algebra, you may assume that = for all special rectangles (the usual proof is to write M as a combination of elementary row operations). The conclusion now follows immediately from the previous subparts. Compareof elementary row operations)....
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