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Unformatted text preview: Real Analysis  Math 630 Homework Set #2  Chapter 3 by Bobby Rohde 91400 Problem 9 & Show that if E is a measurable set, then each translate E + y of E is also measurable. & Proof E is measurable so & A , we have m * A = m *( A & E ) + m *( A & E ). We also know from Problem 7 that outer measure is translation invariant. Thus m *( A + y) = m * A = m *( A & E ) + m *( A & E ) = m *(( A & E ) + y) + m *(( A & E ) + y), & y. m *(( A & E ) + y) + m *(( A & E ) + y) = m *(( A + y) & ( E + y)) + m *(( A + y) & ( E + y)), from the nature of intersection. But any set B may be written as A + y, thus & B , m * B = m *( B & ( E + y)) + m *( B & ( E + y)) = m *( B & ( E + y)) + m *( B & ~( E + y)), since ~( E + y) must equal ( E + y) from the fact that both ~ and + are 11 operations on sets. The last equality thus shows that E + y is measurable, QED. Problem 10 & Show that if E 1 and E 2 are measurable, then m ( E 1 & E 2 ) + m ( E 1 E 2 ) = mE 1 + mE 2 & Proof Since each is measurable we know that mA = m ( A & E 1 ) + m ( A & E 1 & ), mA = m ( A & E 2 ) + m ( A & E 2 & ), A so mE 2 = m ( E 2 & E 1 ) + m ( E 2 & E 1 & ) and mE 1 = m ( E 1 & E 2 ) + m ( E 1 & E 2 & ) Thus mE...
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This note was uploaded on 03/23/2010 for the course MATH 515 taught by Professor Staff during the Spring '08 term at Iowa State.
 Spring '08
 Staff
 Math

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