Math630-1

# Math630-1 - Real Analysis - Math 630 Homework Set #1 by...

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Unformatted text preview: Real Analysis - Math 630 Homework Set #1 by Bobby Rohde 9-7-00 Problem 1 &amp; If A and B are two sets in &amp; with A &amp; B , then mA mB. &amp; Proof Since &amp; is a &amp;-algebra, we know that C = B &amp; A &amp; , with C &amp; A = and C A = B . Hence mB = m ( C A ) = mC + mA mA . mB mA, QED. Problem 2 &amp; Let &lt; E n &gt; be any sequence of sets in &amp; . Then m ( &amp; E n ) mE n . &amp; Proof By Proposition 1.2, a sequence &lt; A n &gt; of sets in &amp; with E n = A n and A n &amp; A m = , n m. Thus m ( E n ) = m ( A n ) = mA n . If we construct A n from E n according to the algorithim used in the proof of Proposition 1.2 then we have A n E n n, and from Problem 1 therefore mA n mE n . Hence we have mA n mE n m ( E n ) mE n . QED. Problem 3 &amp; If there is a set A in &amp; such that mA &lt; &amp; , then m = 0. &amp; Proof By Way of Contradiction (BWOC). Suppose m &amp; = 0, and let mA = &lt; . A &amp; &amp; = &amp; , hence A and &amp; are disjoint. Thus by Property 3, m ( A &amp; ) = mA + m &amp; = + . But A &amp; = A , so m ( A &amp; ) = mA = . Thus = + &gt; . Which is a contradiction, hence m &amp; = 0. QED. Problem 4 &amp; Let nE be &amp; for an infinite set E and equal to the number of elements in E for a finite set. Show that n is a countably additive set function that is translation invariant and defined for all sets of real numbers. translation invariant and defined for all sets of real numbers....
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## Math630-1 - Real Analysis - Math 630 Homework Set #1 by...

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