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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