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Download Brownian Motion and Martingales in Analysis (The Wadsworth by Richard Durrett PDF

By Richard Durrett

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Example text

7 and recall that / 0,2) and / 0 , 2 ) . The set of points of discontinuity of / is {1}, and this set is not v-nu\l (since V is discontinuous at 1) but would become w-null if, in the definition of w-null, one were to replace open intervals by closed intervals or by intervals of the form {a, h]. ]. 8 together suggest that the points at which both / and w fail to be left-continuous or right-continuous will play a role in the characterization of Riemann-Stieltjes integrability of / with respect to w.

The point here is, of course, that if A is a subset of [a, b] which contains, say, the end-point a, then any sequence of open intervals whose union includes A would contain at least one interval whose left end-point lies outside of [a, b] and u would not be defined at such an end-point. 4 to subsets of [a, b] it is necessary to extend mto be a nondecreasing function defined on all of IR. 4. This function u' is left-continuous at a and right-continuous at b, and the next lemma shows that the definition of w-null just given is not as capricious as it might appear to be.

Then there is a point x e la , b] with / (x) = 0. ] (d) Show that if fe J {u ;a ,b ) and if / ( x ) > 0 for all xe[a,fc], then \lfd u > 0 . (e) Show that if u is strictly increasing, if infpS(/;P) = 0, where the infimum is over all partitions P of [a, b], and if ¡ ^ fd u = 0, then every open subinterval of la, ft] contains a point x with /(x) = 0. iSee U. V- Satyanarayana, A note on Riemann-Stieltjes integrals. Am. Math. Monthly, 87 (1980), 477-478. Chapter Four Characterization o f Riemann—Stieltjes Integrability The discussion in the previous chapter left open the question of which bounded functions are Riemann-Stieltjes integrable with respect to a given nondecreas­ ing function over a given closed interval.

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