The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem


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1 The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell Deprtment of Mthemtics, University of Toronto Jnury, BV [, b] Let < b. For f : [, b] R, we define 1 f sup f(t), t [,b] nd if f < we sy tht f is bounded. We define B[, b] to be the set of bounded functions [, b] R, which with the norm is Bnch lgebr. A prtition of [, b] is set P {t 0,..., t n } such tht t 0 < < t n b. For exmple, P {, b} is prtition of [, b]. If Q is prtition of [, b] nd P Q, we sy tht Q is refinement of P. For f : [, b] R, we define V (f, P ) f(t i ) f(t i 1 ). i1 It is strightforwrd to show using the tringle inequlity tht if Q is refinement of P then V (f, P ) V (f, Q). In prticulr, ny prtition P is refinement of {, b}, so f(b) f() V (f, P ). The totl vrition of f : [, b] R is f sup{v (f, P ) : P is prtition of [, b]}, 1 In this note we spek bout functions tht tke vlues in R, becuse this mkes it simpler to tlk bout monotone functions. Once the mchinery is estblished we cn then pply it to the rel nd imginry prts of function tht tkes vlues in C. 1
2 nd if V b f < we sy tht f is of bounded vrition. We denote by BV [, b] the set of functions [, b] R of bounded vrition. For function f BV [, b], we define v : [, b] R by v(x) V x f for x [, b], clled the vrition of f. If f : [, b] R is monotone, it is strightforwrd to check tht V b f f(b) f(), hence tht f is of bounded vrition. We first show tht BV [, b] B[, b]. Lemm 1. If f : [, b] R is of bounded vrition, then f f() + Proof. Let x [, b] If x the result is immedite. If x b, then f(b) f() + f(b) f() f() + Otherwise, P {, x, b} is prtition of [, b] nd f(x) f() V (f, P ) The totl vrition of functions hs severl properties. The following lemm nd tht fct tht functions of bounded vrition re bounded imply tht BV [, b] is n lgebr. Lemm. If f, g BV [, b] nd c R, then the following sttements re true. 1. f 0 if nd only if f is constnt.. (cf) c (f). 3. (f + g) f + g. 4. (fg) f g + g 5. f 6. f V x f + x for x b. Lemm 3. If f : [, b] R is differentible on (, b) nd f <, then f f (b ). Proof. Suppose tht P { t 0 < < t n b} is prtition of [, b]. By the men vlue theorem, for ech j 1,..., n there is some x j (t j 1, t j ) t which f (x j ) f(t j) f(t j 1 ) t j t j 1. N. L. Crothers, Rel Anlysis, p. 04, Lemm 13.3.
3 Then V (f, P ) f(t j ) f(t j 1 ) j1 (t j t j 1 ) f (x j ) j1 f n j1 f (b ). (t j t j 1 ) Lemm 4. If f C 1 [, b], then f f (t) dt. Proof. Let P {t 0,..., t n } be prtition of [, b]. Then, by the fundmentl theorem of clculus, Therefore V (f, P ) f(t j ) f(t j 1 ) j1 tj f (t) t j 1 f (t) dt t j 1 j1 tj j1 f (t) dt. f sup V (f, P ) P f (t) dt. Lemm 5. If f : [, b] R is polynomil, then f f (t) dt. 3
4 Proof. Becuse f is polynomil, f is lso, so f is piecewise monotone, sy f c j f on (t j 1, t j ) for j 1,..., n, for some c j {+1, 1} nd t 0 < < t n b. Then tj t j 1 f (t) dt c j tj t j 1 f (t)dt c j (f(t j ) f(t j 1 )), giving, becuse t 0 < < t n is prtition of [, b], f (t) dt j1 tj f (t) dt t j 1 c j (f(t j ) f(t j 1 )) j1 f(t j ) f(t j 1 ) j1 Lemm 6. If f m is sequence of functions [, b] R tht converges pointwise to some f : [, b] R nd P is some prtition of [, b], then V (f m, P ) V (f, P ). If f m is sequence in BV [, b] tht converges pointwise to some f : [, b] R, then V b f lim inf V b f m. Proof. Sy P {t 0,..., t n }. sequences is liner, Then, becuse tking the limit of convergent lim V (f m, P ) lim j1 j1 f m (t j ) f m (t j 1 ) lim f m(t j ) f m (t j 1 ) f(t j ) f(t j 1 ) j1 V (f, P ). 4
5 Let P {t 0,..., t n } be prtition of [, b]. Then V (f, P ) f(t j ) f(t j 1 ) j1 lim f m(t j ) f m (t j 1 ) j1 lim V (f m, P ) lim inf V b f m. This is true for ny prtition P of [, b], which yields V b f lim inf V b f m. We now prove tht BV [, b] is Bnch spce. 3 Theorem 7. With the norm BV [, b] is Bnch spce. f BV f() + Proof. Using Lemm, it is strightforwrd to check tht BV [, b] is normed liner spce. Suppose tht f m is Cuchy sequence in BV [, b]. By Lemm 1 it follows tht f m is Cuchy sequence in B[, b], nd thus converges in B[, b] to some f B[, b]. Let P be prtition of [, b] nd let ɛ > 0. Becuse f n is Cuchy sequence in BV [, b], there is some N such tht if n, m N then f m f n BV < ɛ. For n N, Lemm 6 yields f f n BV f() f n () + V (f f n, P ) lim ( f m() f n () + V (f m f n, P )) sup ( f m () f n () + V (f m f n, P )) m N sup f m f n BV m N ɛ. Becuse f f N BV [, b] nd f N BV [, b] nd BV [, b] is n lgebr, f (f f N ) + f N BV [, b]. Tht is, the Cuchy sequence f n converges in BV [, b] to f BV [, b], showing tht BV [, b] is complete metric spce nd thus Bnch spce. 3 N. L. Crothers, Rel Anlysis, p. 06, Theorem
6 The following theorem shows tht function of bounded of vrition cn be written s the difference of nondecresing functions. 4 Theorem 8. Let f BV [, b] nd let v be the vrition of f. Then v f nd v re nondecresing. Proof. If x, y [, b], x < y, then, using Lemm, v(y) v(x) V y f V x f V y x f f(y) f(x) f(y) f(x). Tht is, v(y) f(y) v(x) f(x), showing tht v f is nondecresing, nd becuse f is nondecresing we hve f(y) f(x) 0 nd so v(y) v(x) 0. The following theorem tells us tht function of bounded vrition is right or left continuous t point if nd only if its vrition is respectively right or left continuous t the point. 5 Theorem 9. Let f BV [, b] nd let v be the vrition of f. For x [, b], f is right (respectively left) continuous t x if nd only if v is right (respectively left) continuous t x. Proof. Assume tht v is right continuous t x. If ɛ > 0, there is some δ > 0 such tht x y < x+δ implies tht v(y) v(x) v(y) v(x) < ɛ. If x y < x+δ, then f(y) f(x) v(y) v(x) < ɛ, showing tht f is right continuous t x. Assume tht f is right continuous t x, with x < b. Let ɛ > 0. There is some δ > 0 such tht x y < x + δ implies tht f(y) f(x) < ɛ. Becuse x f is supremum over prtitions of [x, b], there is some prtition P {t 0, t 1,..., t n } of [x, b] such tht x f ɛ V (f, P ). Let x y < min{δ, t 1 x}. Then Q {t 0, y, t 1,..., t n } is refinement of P, so Hence x f ɛ V (f, P ) V (f, Q) f(y) f(t 0 ) + V (f, {y, t 1,..., t n }) < ɛ + y f. ɛ > x f y f V y x f v(y) v(x) v(y) v(x), showing tht v is right continuous t x. 4 N. L. Crothers, Rel Anlysis, p. 07, Theorem N. L. Crothers, Rel Anlysis, p. 07, Theorem
7 For f BV [, b] nd for v the vrition of f, we define the positive vrition of f s v(x) + f(x) f() p(x), x [, b], nd the negtive vrition of f s n(x) v(x) f(x) + f(), x [, b]. We cn write the vrition s v p + n. We now estblish properties of the positive nd negtive vritions. 6 Theorem 10. Let f BV [, b], let v be its vrition, let p be its positive vrition, nd let n be its negtive vrition. Then 0 p v nd 0 n v, nd p nd n re nondecresing. Proof. For x [, b], v(x) V x f f(x) f(). Becuse v(x) (f(x) f()), we hve p(x) 0, nd becuse v(x) f(x) f() we hve n(x) 0. And then v p + n implies tht p v nd n v. For x < y, nd p(y) p(x) v(y) + f(y) v(x) f(x) 1 (V y x f + (f(y) f(x))) 1 ( f(y) f(x) + (f(y) f(x))) 0 n(y) n(x) v(y) f(y) v(x) + f(x) 1 (V y x f (f(y) f(x))) 1 ( f(y) f(x) (f(y) f(x))) 0. We now prove tht BV [, b] is Bnch lgebr. 7 Theorem 11. BV [, b] is Bnch lgebr. 6 N. L. Crothers, Rel Anlysis, p. 09, Proposition N. L. Crothers, Rel Anlysis, p. 09, Proposition
8 Proof. For f 1, f BV [, b], let v 1, v, p 1, p, n 1, n be their vritions, positive vritions, nd negtive vritions, respectively. Then f 1 f (f 1 () + p 1 n 1 )(f () + p n ) f 1 ()f () + p 1 p + n 1 n n 1 p n p 1 + f 1 ()p + f ()p 1 f 1 ()n f ()n 1. Using this nd the fct tht if f is nondecresing then f f(b) f(), f 1 f BV f 1 () f () + (f 1 f ) f 1 () f () + (p 1 p ) + (n 1 n ) + (n 1 p ) + (n p 1 ) + f 1 () p + f () p 1 + f 1 () n + f () n 1 f 1 () f () + p 1 (b)p (b) + n 1 (b)n (b) + n 1 (b)p (b) + n (b)p 1 (b) + f 1 () p (b) + f () p 1 (b) + f 1 () n (b) + f () n 1 (b) ( f 1 () + p 1 (b) + n 1 (b))( f () + p (b) + n (b)) ( f 1 () + v 1 (b))( f () + v (b)) f 1 BV f BV, which shows tht BV [, b] is normed lgebr. And BV [, b] is Bnch spce, so BV [, b] is Bnch lgebr. Theorem 1. If f C 1 [, b], then f f (t) dt. Let (f ) + nd (f ) be the positive nd negtive prts of f nd let p nd n be the positive nd negtive vritions of f. Then, for x [, b], p(x) x (f ) + (t)dt, n(x) x (f ) (t)dt. Proof. Lemm 4 sttes tht V b f (t) dt. Becuse f is continuous it is Riemnn integrble, hence for ny ɛ > 0 there is some prtition P {t 0,..., t n } of [, b] such tht if x j [t j 1, t j ] for j 1,..., n then f (t) dt f (x j ) (t j t j 1 ) < ɛ. j1 By the men vlue theorem, for ech j 1,..., n there is some x j (t j 1, t j ) such tht f (x j ) f(tj) f(tj 1) t j t j 1. Then V (f, P ) f(t j ) f(t j 1 ) j1 f (x j ) (t j t j 1 ), j1 8
9 so nd thus This is true for ll ɛ > 0, therefore f (t) dt V (f, P ) < ɛ, f (t) dt < V (f, P ) + ɛ f + ɛ. which is wht we wnted to show. Write f (t) dt f, g(t) (f ) + (t) mx{f (t), 0}, h(t) (f ) (t) min{f (t), 0}. These stisfy g + h f nd g h f. Using the fundmentl theorem of clculus, nd p(x) 1 (v(x) + f(x) f()) 1 ( x ) V x f + f (t)dt ( 1 b ) b f (t) dt + f (t)dt g(t)dt n(x) 1 (v(x) f(x) + f()) 1 ( x ) V x f f (t)dt 1 ( x x ) f (t) dt f (t)dt x h(t)dt. Helly s selection theorem We will use the following lemms in the proof of the Helly selection theorem. 8 8 N. L. Crothers, Rel Anlysis, p. 10, Theorem 13.13; p. 11, Lemm 13.14; p. 11, Lemm
10 Lemm 13. Suppose tht X is set, tht f n : X R is sequence of functions, nd tht there is some K such tht f n K for ll n. If D is countble subset of X, then there is subsequence of f n tht converges pointwise on D to some φ : D R, which stisfies φ K. Proof. Sy D {x k : k 1}. Write fn 0 f n. The sequence of rel numbers fn(x 0 1 ) stisfies fn(x 0 1 ) [ K, K] for ll n, nd since the set [ K, K] is compct there is subsequence fn(x 1 1 ) of fn(x 0 1 ) tht converges, sy to φ(x 1 ) [ K, K]. Suppose tht fn m (x m ) is subsequence of fn m 1 (x m ) tht converges to φ(x m ) [ K, K]. Then the sequence of rel numbers fn m (x m+1 ) stisfies fn m (x m+1 ) [ K, K] for ll n, nd so there is subsequence fn m+1 (x m+1 ) of fn m (x m+1 ) tht converges, sy to φ(x m+1 ) [ K, K]. Let k 1. Then one checks tht fn n (x k ) φ(x k ) s n, nmely, fn n is subsequence of f n tht converges pointwise on D to φ, nd for ech k we hve φ(x k ) [ K, K]. Lemm 14. Let D [, b] with D nd b sup D. nondecresing, then Φ : [, b] R defined by Φ(x) sup{φ(t) : t [, x] D} is nondecresing nd the restriction of Φ to D is equl to φ. If φ : D R is Lemm 15. If f n : [, b] R is sequence of nondecresing functions nd there is some K such tht f n K for ll n, then there is nondecresing function f : [, b] R, stisfying f K, nd subsequence of f n tht converges pointwise to f. Proof. Let D (Q [, b]) {}. By Lemm 13, there is function φ : D R nd subsequence f n of f n tht converges pointwise on D to φ, nd φ K. Becuse ech f n is nondecresing, if x, y D nd x < y then φ(x) lim n f n (x) lim n f n (y) φ(y), nmely, φ is nondecresing. D is dense subset of [, b] nd D, so pplying Lemm 14, there is nondecresing function Φ : [, b] R such tht for x D, Φ(x) φ(x) lim n f n (x). Suppose tht Φ is continuous t x [, b] nd let ɛ > 0. Using the fct tht Φ is continuous t x, there re p, q Q [, b] such tht p < x < q nd Φ(q) Φ(p) Φ(q) Φ(p) < ɛ. Becuse p, q D, f n (p) Φ(p) nd f n (q) Φ(q), so there is some N such tht n N implies tht both f n (p) Φ(p) < ɛ nd f n (q) Φ(q) < ɛ. Then for n N, becuse ech function f n is nondecresing, f n (x) f n (p) Φ(p) ɛ Φ(q) ɛ Φ(x) ɛ. 10
11 Likewise, for n N, f n (x) f n (q) Φ(q) + ɛ < Φ(p) + ɛ Φ(x) + ɛ. This shows tht if Φ is continuous t x [, b] then f n (x) Φ(x). Let D(Φ) be the collection of those x [, b] such tht Φ is not continuous t x. Becuse Φ is monotone, D(Φ) is countble. So we hve estblished tht if x [, b] \ D(Φ) then f n (x) Φ(x). Becuse f n : [, b] R stisfies f n K nd D(Φ) is countble, Lemm 13 tells us tht there is function F : D R nd subsequence f bn of f n such tht f bn converges pointwise on D to F, nd F K. We define f : [, b] R by f(x) Φ(x) for x D(Φ) nd f(x) F (x) for x D(Φ). f K. For x D(Φ), f n (x) converges to Φ(x) f(x), nd f bn (x) is subsequence of f n (x) so f bn (x) converges to f(x). For x D(Φ), f bn (x) converges to F (x) f(x). Therefore, for ny x [, b] we hve tht f bn (x) f(x), nmely, f bn converges pointwise to f. Becuse ech function f bn is nondecresing, it follows tht f is nondecresing. Finlly we prove the pointwise Helly selection theorem. 9 Theorem 16. Let f n be sequence in BV [, b] nd suppose there is some K with f n BV K for ll n. There is some subsequence of f n tht converges pointwise to some f BV [, b], stisfying f BV K. Proof. Let v n be the vrition of f n. This stisfies, for ny n, nd v n f n K v n f n v n + f n K + f n BV K. Theorem 8 tells us tht v n f n nd v n re nondecresing, so we cn pply Lemm 15 to get tht there is nondecresing function g : [, b] R nd subsequence v n f n of v n f n tht converges pointwise to g. Then we use Lemm 15 gin to get tht there is nondecresing function h : [, b] R nd subsequence v bn of v n tht converges pointwise to h. Becuse g nd h re pointwise limits of nondecresing functions, they re ech nondecresing nd so belong to BV [, b]. We define f h g BV [, b]. For x [, b], lim f b n n (x) lim v b n n (x) lim (v b n n (x) f bn (x)) h(x) g(x) f(x), 9 N. L. Crothers, Rel Anlysis, p. 1, Theorem
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