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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online