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Math630-2

# Math630-2 - Real Analysis Math 630 Homework Set#2 Chapter 3...

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Unformatted text preview: Real Analysis - Math 630 Homework Set #2 - Chapter 3 by Bobby Rohde 9-14-00 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 1-1 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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Math630-2 - Real Analysis Math 630 Homework Set#2 Chapter 3...

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