By S. L. Loney

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Integrating t h e Pk{t) gives k Pk(t) = f p {r)dT k J—ο which is a continuous function, monotonely increasing from 0 t o 1. Its classical derivative P' {t) is defined almost everywhere a n d equal t o ρ*(ί). T h e pointwise limit P(t) = lim P (t) k k k—¥oo is a well defined, continuous, monotone increasing function (sometimes called the 'Devil's staircase') which h a s a classical derivative defined a n d equal t o zero almost everywhere. We can give a n explicit definition of P(t) as follows: +00 +00 _1 For t = Σ j f c £ * ( l ~ξ)€Ε define P{t) = Σ J * / 2 * .

Gen. 1 3 , 3 1 , (1980). E . A. Tough, Adv. Phys. 37,471,(1988). E. J a k e m a n , J. Opt. A. Pure Appl. Opt. 1, 784,(1999). 7. M. A. Stegun, Handbook of Mathematical Ed. (Dover, New York, 1970). Functions, 9th 8. V. Berry, Philos. Trans. R. Soc. London, Ser. A 2 7 3 , 6 1 1 , (1973). 9. W . Feller, An Introduction to Probability and Its Applications, Vols I & II. 2nd Edition, (John Wiley & Sons, London & New York, 1971). M. Ding & W. Yang, Phys. Rev. E, 5 2 , 207, (1995). 10. M. Evans, N. Hastings & B .

E. any set, finite or infinite, which contains no limit point) then it constitutes a set of Lebesgue measure zero, and a mass distribution concentrated on A can be defined by assigning an appropriately weighted Dirac measure t o each point of A. This does not exhaust the possible types of singular measure as the example of t h e Cantor set makes plain. 2 Cantor sets For the sake of clarity we recall briefly the usual construction of a Cantor subset of t h e interval [0,1]. First, divide t h e interval [0,1] into three p a r t s of lengths proportional t o ξ, 1 - 2ξ, and ξ respectively, where 0 < ξ < 1/2.