On the Connection between the Hilger and Radon–Nikodym Derivatives
نویسندگان
چکیده
We show that the Hilger derivative on time scales is a special case of the Radon–Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.
منابع مشابه
Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon Nikodym Property
In this paper we prove the differentiability of Lipschitz maps X → V , where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direct...
متن کاملThe idempotent Radon--Nikodym theorem has a converse statement
Idempotent integration is an analogue of the Lebesgue integration where σ-additive measures are replaced by σ-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon–Nikodym derivatives, which turns out to be crucial in all of these applications, was prov...
متن کاملRadon–Nikodym derivatives of quantum operations
Given a completely positive ~CP! map T, there is a theorem of the Radon– Nikodym type @W. B. Arveson, Acta Math. 123, 141 ~1969!; V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 ~1986!# that completely characterizes all CP maps S such that T2S is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon–Nikodym formali...
متن کاملInformation-Theoretic Demensionality Reduction for Poisson Models: Supplementary Material
Proof of Theorem 1. We first establish the following Lemma. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivatives of probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f Y is the Radon-Nikodym derivatives of probability measure P θ Y with respect to QY provided that P θ Y QY . Note that i...
متن کاملA new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011