160 1. Measure theory (i) If H : R R is the Heaviside function H := 1[0,+∞), show that μH is equal to the Dirac measure δ0 at the origin (defined in Example 1.4.22). (ii) If F = n cnJn is a jump function (as defined in Definition 1.6.30), show that μF is equal to the linear combination cnδxn of delta functions (as defined in Exercise 1.4.22), where xn is the point of discontinuity for the basic jump function Jn. Exercise 1.7.15 (Lebesgue-Stieltjes measure, singular continuous case). (i) If F : R R is a monotone non-decreasing function, show that F is continuous if and only if μF ({x}) = 0 for all x R. (ii) If F is the Cantor function (defined in Exercise 1.6.48), show that μF is a probability measure supported on the middle-thirds Can- tor set (see Exercise 1.2.9) in the sense that μF (R\C) = 0. The measure μF is known as Cantor measure. (iii) If μF is Cantor measure, establish the self-similarity properties μ( 1 3 · E) = 1 2 μ(E) and μ( 1 3 · E + 2 3 ) = 1 2 μ(E) for every Borel- measurable E [0, 1], where 1 3 · E := { 1 3 x : x E}. Exercise 1.7.16 (Connection with Riemann-Stieltjes integral). Let F : R→ R be monotone non-decreasing, let [a, b] be a compact interval, and let f : [a, b] R be continuous. Suppose that F is continuous at the endpoints a, b of the interval. Show that for every ε 0 there exists δ 0 such that | n i=1 f(ti ∗)(F (ti) F (ti−1)) [a,b] f dF | ε whenever a = t0 t1 . . . tn = b and ti [ti−1,ti] for 1 i n are such that sup1≤i≤n |ti ti−1| δ. In the language of the Riemann-Stieltjes integral, this result asserts that the Lebesgue-Stieltjes integral extends the Riemann-Stieltjes integral. Exercise 1.7.17 (Integration by parts formula). Let F, G: R R be monotone non-decreasing and continuous. Show that [a,b] F dG = [a,b] G dF + F (b)G(b) F (a)G(a) for any compact interval [a, b]. (Hint: Use Exercise 1.7.16.) This formula can be partially extended to the case when one or both of F, G have dis- continuities, but care must be taken when F and G are simultaneously discontinuous at the same location.
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