Singular Continuous Measures by Michael Pejic 5/14/10

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1 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable itersectios as well. A measure space is a triplet X, A, µ, where X is a set, A is a σ-algebra o X ad the measure µ : A [0, ] satisfies: i µ = 0; ii if {A } J A is a coutable collectio of disoit sets, the µ A = µ A J J If µ X is fiite, µ is termed a fiite measure; if X ca be expressed as the coutable uio of sets i A, each of which has fiite measure, the µ is termed a σ-fiite measure. The support of a measure is the largest closed set such that ay ope subset of it has ozero measure. A fuctio f : X [, ] is A-measurable if the preimage f c, ] is i A for all c R hece the preimage of ay Borel set is i A. A simple A-measurable fuctio is oe takig o oly fiitely may distict values, all of which are fiite. Such a fuctio is itegrable if its support has fiite measure. The itegral of a positive A-measurable fuctio with respect to measure µ is defied as { } f dµ = sup a µ A itegrable simple fuctios a χ A f poitwise The itegral of a geeral A-measurable fuctio with respect to measure µ is defied by f dµ = f >0 dµ f <0 dµ Note that this excludes ay otio of coditioal covergece. Fially, a property is said to hold almost everywhere with respect to µ if it fails o a set of measure zero. Examples i The trivial measure which assigs measure 0 to every elemet of the σ- algebra.

2 ii As a example of fiite measures, probability measures Ω, E, π with π Ω =. iii As a example of a σ-fiite measure, Lebesgue measure with X = R, A give by Borel sets, ad measure λ give by λ A = if b a all coutable collectios of ope itervals such that a, b A The cotiuous fuctios f : [a, b] R are measurable ad for them the defiitio of itegratio agrees with the stadard Riema itegral, b f x dx = f χ [a,b] dλ = f dλ a Stieltes Itegral For X = R ad A give by Borel sets, there is a equivalet formulatio. Let ϕ : R be odecreasig ad cotiuous from the left. Let the Lebesgue- Stieltes measure µ ϕ be the measure iduced by µ ϕ [a, b = ϕ b ϕ a. Coversely, give ay measure o Borel subsets of R that is fiite o bouded sets, µ, defie ϕ : R R by µ [x, 0 if x < 0 ϕ x = 0 if x = 0 µ [0, x if x > 0 The µ = µ ϕ. For cotiuous fuctios f : [a, b] R, this agrees with the stadard Riema-Stieltes itegral, b f dµ ϕ = f x dϕ x = f x ϕ x+ ϕ x [a,b] a lim max k x k+ x k 0 =0 where the limit is take over all partitios a = x 0 x 0 x... x = b. Examples i For ϕ x = x, we recover the Lebesgue measure, µ ϕ = λ. ii For ϕ = Θ, the Heaviside fuctio { 0 if x 0 Θ x = if x > 0 [a,b]

3 we get a measure µ Θ that has a atom of mass at x = 0, which is typically for those igorat of measure theory deoted dϕ = δ x dx for δ the Dirac delta fuctio. Note this measure, ulike Lebesgue measure, ca be exteded from the σ-algebra of Borel sets to the σ-algebra of all subsets of R. Note that discotiuities i ϕ correspod to atoms with mass give by the height of the ump. Absolute Cotiuity Give two A-measures, µ, ν, if a set is of measure 0 with respect to ν wheever it is measure 0 with respect to µ, the ν is said to be absolutely cotiuous with respect to µ, deoted ν µ. If µ, ν are i additio σ-fiite, by the Rado-Nikodym theorem there is a A-measurable fuctio, deoted dν dµ, such that for ay A A, ν A = A dν dµ dµ This fuctio is uique withi a µ-ull fuctio. Mutual Sigularity Give two A-measures, µ, ν, if there are disoit sets E, F such that for ay set A A, A E, A F A ad µ A E = µ A, ν A F = ν A, the µ, ν are said to be mutually sigular, deoted µ ν. Decompositio of a Measure For two σ-fiite, A-measures µ, ν, by the Lebesgue decompostio theorem µ = µ ac + µ sig, where µ ac ν ad µ sig ν. The the Rado-Nikodym derivative dµ dν is defied to be dµac dν. Example Take λ ad µ Θ. These are mutually sigular sice for ay Borel set A, λ A = λ A\ {0} whereas µ Θ A = µ Θ A {0}. Specializig to the case where oe of the measures is Lebesgue measure, there is a further decompositio of the sigular part ito atoms ad a sigular cotiuous part. The first is familiar from physical cocepts of poit masses or charges; the secod is far less so.

4 First Example of Sigular Cotiuous Measure: Devil s Staircase Cosider the sequece of o-decreasig, cotiuous fuctios f =0 from the uit iterval oto itself with f [, ] = f [ 9, 9] = 4, f [, ] =, f [ 7 9, 8 9] = 4 f [ 7, 7] = 8, f [ 9, 9] = 4, f [ 7 7, 8 7] = 8, f [, ] =, f [ 9 7, 0 7] = 5 8, f [ 7 9, 8 9] = 4, f [ 5 7, 6 7] = 7 8. with iterveig values determied by liear iterpolatio. This coverges i supremum orm to a o-decreasig, cotiuous fuctio f : [0, ] [0, ] that clearly has derivative 0 almost everywhere with respect to Lebesgue measure sice the Lebesgue measure of the middle-third Cator set C is 0, so the Rado-Nikodym derivative dµ f = 0, yet µ dλ f is certaily ot 0 sice µ f [0, ] =. At the same time, sice f is cotiuous, µ f cotais o atoms. Therefore, µ f is purely sigular cotiuous. Note µ f is equal to the Hausdorff measure h log defied by h log A = lim ε 0 + if diam B log B B ope covers B of A by balls of diameter less tha ε Doig itegrals with respect to µ f by takig the limit as of doig them with µ f is tedious. I some cases, this ca be made easier by iterpretig the graph of f as a affie iterated fractal system with base B = 0, 0,, 0, 0, ad geerators G = 0, 0,, 0, 0,, G =,,,,,, ad G =,,,,, with correspodig affie trasformatios A x, y = x, y A x, y = x +, 4

5 A x, y = x +, y + so graph f = A graph f A graph f A graph f, which implies for x [0, ], x, f x graph f x +, graph f x +, f x + graph f which i tur imply f x if x [ ] 0, f x = if x [, ] f x + if x [, ] dµ f x if x [ ] 0, dµ f x = 0 if x [, ] dµ f x if x [, ] This ca be used to fid the momets: m = x dµ f x = [0, x dµ f x + [, x dµ f x = [0, x dµ f x + [, x dµ f x x x + dµ f x so for, = dµf x + = m + m + m = = = m m 5

6 Usig this ad m 0 = 0, the first few are give by m =, m =, m 8 = 5, m 6 4 = Secod Example of Sigular Cotiuous Measure: Ferec Riesz The previous example is ot really that differet from the atomic case except the support of the sigular part of the measure is a ucoutable fractal set rather tha a coutable collectio of poits. To see how strage sigular measures ca be, cosider the followig costructio due to Ferec Riesz: Pick a t 0,. Start with the graph of F 0 beig the lie segmet from 0, 0 to,. To costruct the graph of F +, replace each lie segmet x 0, y 0 x, y i the graph of F with the pair of lie segmets x0 + x x 0, y 0, t y t y ad x0 + x, t y t y x, y I the limit as, the sequece of icreasig, cotiuous fuctios F =0 from the uit iterval oto itself coverges i sup orm to a odecreasig actually icreasig, cotiuous fuctio F : [0, ] [0, ]. This ca also be iterpreted as the affie iterated fractal system give by base B = 0, 0,, 0, 0, ad geerators G = 0, 0,, 0, 0, +t ad G =, +t,, +t,, with correspodig affie trasformatios A x, y = x, + t y ad A x, y = x +, t y + + t By repeated applicatio of the chai rule, for x =.b b b... i biary { df + t if b = 0 dx = x t if b = =0 At biary fractios, the two differet biary expasios correspod to takig the respective oe-sided derivatives. If lim sup umber of 0 s i first biary digits of x 6 < log t log + t log t

7 the df dx x exists ad is equal to 0. Yet, by applyig the cetral limit theorem to the biomial distributio it ca be show see below for details that, with respect to Lebesgue measure, for almost every x [0, ], so df dx umber of 0 s i first biary digits of x lim = = 0 almost everywhere, so the Rado-Nikodym derivative dµ F dλ = 0. Sice F is cotiuous, µ F has o atoms, so it is purely sigular cotiuous. Ulike i the previous example, however, the support of µ F is all of [0, ] sice those x with lim sup umber of 0 s i first biary digits of x log t log + t log t are a dese subset; hece, F is icreasig. The formulatio as a iterative fractal system ca also be used to do certai itegrals with respect to µ F. Sice graph F = A graph F A graph F which implies for x [0, ], x, + t F x graph F ad x +, t F x + + t graph F which i tur imply we have F x = dµ F x = { +t F x if x [ 0, +t F x + if x [, ] t { +t dµ F x if x [ 0, t dµ F x if x [, ] Usig this, we ca calcute the momets by gettig a relatio, m = 0 x dµ F x dx = 0 x + t dµ F x + 7 ] ] x t dµ F x

8 = + t The for 0 x dµf x + t m = + t m + + t m + + m = t + = Usig m 0 =, the first few are give by: m = t, m = t + t m 4 = t 0 x + dµ F x = m, m = t m + + t t 5t ad for t = : m = 4, m = 8, m = 7 4, m 4 = t t 7 4 Proof of Assertio I order to show that, with respect to Lebesgue measure, for almost every x [0, ], umber of 0 s i first biary digits of x lim it is oly ecessary to show that for ay p, ], the set of x [0, ] such that lim sup umber of 0 s i first biary digits of x has Lebesgue measure 0 sice by symmetry betwee 0 s ad s i the biary expasio, the correspodig set of x [0, ] such that lim if umber of 0 s i first biary digits of x = p p will also have Lebesgue measure 0, ad the, sice coutable uios of measure 0 sets are of measure 0, we ca simply apply the result to the sequece + for the values of p. = 8

9 Now take ay δ > 0 ad let q, p. The the proportio of biary fractios with deomiator that have proportio of 0 s i first biary digits greater tha or equal to q is give by = q For large, by applyig the cetral limit theorem to the biomial distributio, for of order from, π 4 e 4 = π e sice for = the mea is ad the variace. The same result also 4 follows usig Stirlig s formula. Ufortuately, we are iterested i of order from ; however, by umerical computatio the Gaussia is actually a rather drastic overestimate i this regio, so = q lettig x = = < q = q π e t < q π e dt e x dx < e x dx π q π t. Therefore, for sufficietly large, = q < erfc q e q π q from the asymptotic form for the complimetary error fuctio. Hece, there is a sequece of umbers, a =0 with the followig properties: i the sequece has limit ; ii the sum a =0 = q 9

10 is fiite. For example, from the precedig asymptotic expressio, a = r for r, e q will certaily work by the limit compariso test. Let the sum be deoted by M. Now cosider the subset B give by the uio of balls of diameter aδ M cetered at biary fractios with deomiator that have proportio of 0 s i their first biary digits greater tha or equal to q. The Lebesgue measure of B is less tha or equal to =0 a δ M = q = δ Sice a, there is a N such that for all m > N, am δ M umber such that lim sup umber of 0 s i first biary digits of x so by defiitio there is a > N such that >. Let x be ay p umber of 0 s i first biary digits of x q The x is withi < aδ of a biary fractio with deomiator M that has proportio of 0 s i first biary digits greater tha or equal to q, so x is i B. Sice x was arbitrary, all such umbers are i B. Sice δ was arbitrary, we ca coclude that the set of all such umbers is of Lebesgue measure 0. 0