Unique nonnegative radon measure

Reference no: EM13990805

Recall that a Radon measure on R is a function µ into R, defined on all bounded Borel sets and countably additive on the Borel subsets of any fixed bounded interval. Show that for any convex function f on an open interval U ⊂ R, there is a unique nonnegative Radon measure µ such that f 1(y+) - f 1(x +) = µ((x, y]) and f 1(y-) - f 1(x -) = µ([x, y)) for all x y in U.

Reference no: EM13990805

Questions Cloud

Continuous with respect to counting measure : Lebesgue measure λ is absolutely continuous with respect to counting measure c on [0, 1], which is not σ-finite. Show that the conclusion of the Radon-Nikodym theorem does not hold in this case (there is no "derivative" dλ/dc having the properties..

Problem regarding the orthonormal basis : Let (X, S, µ) and (Y, T , ν) be two measure spaces. Let { fn } be an orthonormal basis of L2(X, S, µ) and {gn } an orthonormal basis of L2(Y, T , ν). Show that the set of all functions hmn (x, y) := fm (x )gn (y) is an orthonormal basis of L2(X × ..

Discuss and predict the outcomes with the use of panel data : The reading for this module discusses panel data and the use of pooled cross-sectional analysis. Post a research question you would be interested in pursuing. Then discuss and predict the outcomes with the use of panel data and the use of pooled c..

Prepare necessary adjusting entries for month of september : On September 30, Sweet Cheeks completed service for which it had received payment in August, $1,500. Prepare all necessary adjusting entries for the month of September.

Unique nonnegative radon measure : Recall that a Radon measure on R is a function µ into R, defined on all bounded Borel sets and countably additive on the Borel subsets of any fixed bounded interval.

Project breakdown risks and staffing : Project Breakdown, Risks and Staffing - PMI is one of the world's largest not-for-profit membership associations for the project management profession

How can global city accommodate both global & local identity : Explain two or three consequences that may occur if a global city exiled the middle and working class?

Convex function defined on a convex : Let f be a convex function defined on a convex set A ⊂ Rk. For some fixed x and y in Rk let g(t ) := f (x + ty) + f (x - ty) whenever this is defined. Show that g is a nondecreasing function defined on an interval (possibly empty) in R.

Problem regarding the convex function : 1. For any set F and point x in a metric space recall that d(x, F ) := inf{d(x, y): y ∈ F }. Let F be a closed set in a normed linear space S with the usual distance d(x, y) := x - y . Show that d(·, F ) isa convex funct..

User Account

All Pages