1.7. Outer measures, pre-measures, product measures 159 on R, show that there exists a monotone function F : R → R such that μ = μF . Radon measures are studied in more detail in §1.10 of An epsilon of room, Vol. I. Exercise 1.7.12 (Near uniqueness). If F, F : R → R are monotone non- decreasing functions, show that μF = μF if and only if there exists a con- stant C ∈ R such that F+(x) = F+(x) + C and F−(x) = F−(x) + C for all x ∈ R. Note that this implies that the value of F at its points of discon- tinuity are irrelevant for the purposes of determining the Lebesgue-Stieltjes measure μF in particular, μF = μF+ = μF− . In the special case when F+(−∞) = 0 and F−(+∞) = 1, then μF is a probability measure, and F+(x) = μF ((−∞,x]) is known as the cumulative distribution function of μF . Now we give some examples of Lebesgue-Stieltjes measure. Exercise 1.7.13 (Lebesgue-Stieltjes measure, absolutely continuous case). (i) If F : R → R is the identity function F (x) = x, show that μF is equal to Lebesgue measure m. (ii) If F : R → R is monotone non-decreasing and absolutely continu- ous (which, in particular, implies that F exists and is absolutely integrable, show that μF = mF in the sense of Exercise 1.4.48, thus μF (E) = E F (x) dx for any Borel measurable E, and R f(x) dμF (x) = R f(x)F (x) dx for any unsigned Borel measurable f : R → [0, +∞]. In view of the above exercise, the integral R f dμF is often abbreviated R f dF , and referred to as the Lebesgue-Stieltjes integral of f with respect to F . In particular, observe the identity [a,b] dF = F+(b) − F−(a) for any monotone non-decreasing F : R → R and any −∞ b a +∞, which can be viewed as yet another formulation of the fundamental theorem of calculus. Exercise 1.7.14 (Lebesgue-Stieltjes measure, pure point case).

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.