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Royden_3

# Royden_3 - Chapter 3 Lebesgue Measure Written by Men-Gen...

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Chapter 3: Lebesgue Measure Written by Men-Gen Tsai email: [email protected] 1. Let m be a countably additive measure defined for all sets in a σ -algebra M . If A and B are two sets in M with A B , then mA mB . This property is called monotonicity . Proof: B = A ( A - B ). A and A - B are disjoint. Since m is a countably additive measure, mB = mA + m ( A - B ). Note that m is nonnegative, and m ( A - B ) 0. Hence mA mB . 2. Let < E n > be any sequence of sets in M . Then m ( E n ) mE n . [Hint: Use Proposition 1.2.] This property of a measure is called count- able subadditivity . Proof: By Proposition 1.2 on page 17, since M is a σ -algebra, there is a sequence < B n > of sets in M such that B n B m = φ for n = m and i =1 B i = i =1 E i . Since m is a countably additive measure and B i E i for all i , by Problem 3.1 I have that m ( E n ) = m ( B n ) = mB n mE n . 3. If there is a set A in M such that mA < , then = 0. Proof: Note that A = A φ and A and φ are disjoint, and thus mA = mA + mφ. Since mA < , = 0 precisely. 1

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4. 5. Let A be the set of rational numbers between 0 and 1, and let { I n } be a finite collection of open intervals covering A . Then l ( I n ) 1. Proof 1: (due to Meng-Gen Tsai) Since 0 A , there is an open interval J 1 in { I n } such that 0 J 1 . Let J 1 = ( a 1 , b 1 ). Note that a 1 < 0 and b 1 > 0. If b 1 1, then l ( I n ) l ( J 1 ) = b 1 - a 1 1 . Suppose not. If a 1 is rational, then I can find an open interval J 2 ∈ { I n } such that a 1 J 2 . If a 1 is irrational, I consider the following cases.
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Royden_3 - Chapter 3 Lebesgue Measure Written by Men-Gen...

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