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 = φ be a set, M a σ-algebra on X and let μ,ν be two positive measures on M . (a) Let A ⊆ 2 X be an algebra such that μ ( A ) = ν ( A ) for all A ∈ A . If μ ( X ) = ν ( X ) < ∞ , then show that μ ( A ) = ν ( A ) for all A ∈ σ ( A ). (b) Show that the previous subpart need not be true if μ ( X ) = ν ( X ) = ∞ . (c) Show that the conclusion of part (a) holds if X = ∪ A i , with A i ∈ A , and μ ( A i ) = ν ( A i ) < ∞ . (d) Let M be an n × n matrix and A ∈ L ( R n ), show that λ ( MA ) = | det M | λ ( A ). [ Hint: First assume M is invertible. Define μ ( A ) = | det M | λ ( M- 1 A ), and verify μ is a measure. From standard linear algebra, you may assume that μ = λ 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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This note was uploaded on 07/10/2009 for the course DSD 11 taught by Professor Dsfdfs during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.
- Spring '09