ma108-2 - MA108 Homework 2 Seung Woo Shin January 25th 2009...

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Unformatted text preview: MA108 Homework 2 Seung Woo Shin January 25th 2009 1 First, note that by Theorem 13.9, v ( x ) = V x a f is right continuous on [ a,b ]. Fix > 0 and let P = a = x < x 1 < x 2 < ... < x n = b . Then, obviously there is δ > 0 such that V y x < / 2 n whenever | x- y | < δ . Now, we construct another partition Q using P as follows. Q = { y ,...,y 2 n } where y 2 k = x k and y 2 k +1 ∈ ( y 2 k ,y 2 k + δ ). Now, construct a continuous f ( x ) such that f ( x ) = sgn( α ( y 2 k )- α ( y 2 k- 1 )) when x ∈ [ y 2 k- 1 ,y 2 k ] and || f || ∞ ≤ 1 (it is obviously possible). Then, Z b a fdα- V ( α,Q ) = 2 n X i =1 ( Z y i y i- 1 fdα- V ( α,Q )) = n X k =1 sgn( α ( y 2 k )- α ( y 2 k- 1 ))( α ( y 2 k )- α ( y 2 k- 1 )) + n- 1 X k =0 Z y 2 k y 2 k +1 fdα- V ( α,Q ) = n X k =1 | α ( y 2 k )- α ( y 2 k- 1 ) | + n- 1 X k =0 Z y 2 k y 2 k +1 fdα- V ( α,Q ) =- n- 1 X k =0 | α ( y 2 k +1 )- α ( y 2 k ) | + n- 1 X k =0 Z y 2 k y 2 k +1 fdα >- n / 2 n- n / 2 n =- (Note that | α ( y 2 k +1 )- α ( y 2 k ) | < / 2 n since y 2 k +1 ∈ ( y 2 k ,y 2 k + δ ) and that by Theorem 14.16, | R y 2 k y 2 k +1 fdα | ≤ || f || ∞ V y 2 k y 2 k +1 α ≤ V y 2 k y 2 k +1 α < / 2 n ) Thus, we have Z b a fdα ≥ V ( α,Q )- 1 Since Q ⊃ P , V ( α,Q ) ≥ V ( α,P ). So, finally, Z b a fdα ≥ V ( α,P )- Now, note that by Corollary 14.13, C [ a,b ] ⊂ R α [ a,b ]. Suppose P = { x ,...,x n } and T = { t 1 ,...,t n } is an appropriate selection of points for P . Then, for any f such that || f || ∞ ≤ 1, S ( f,P,T ) ≤ n X i =1 | α ( x i )- α ( x i- 1 ) | ≤ V b a ( α ) So, sup { R b a fdα : || f || ∞ ≤ 1 } ≤ V b a ( α ) = sup V ( α,P ) ≤ sup { R b a fdα : || f || ∞ ≤ 1 } . Hence, V b a ( α ) = sup { Z b a fdα : || f || ∞ ≤ 1 } 2 Let β = α ◦ φ . I’ll first show that β ∈ BV [ c,d ]. Let P be any partition of [ c,d ]. Then, if we let Q = φ ( P ), V ( β,P ) = V ( α,Q ) since φ is strictly increasing. So, β ∈ BV [ c,d ].We want to show that g ∈ R β [ c,d ]. That is, ∃ I ∈ R such that ∀ > 0, there is a partition P * of [ c,d ] such that | S ( g,P,T )- I | < for all refinements P ⊃ P * and all selections of points T ....
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ma108-2 - MA108 Homework 2 Seung Woo Shin January 25th 2009...

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