Copyright - The Theory of Lebesgue Measure and IntegrationELSEVIERTheory of Lebesgue Measure & Integration
\bf Definition\ 4.2\ (Integral\ of\ nonnegative\ measurable\ function) Let E\in\mathscr M and f:E\longrightarrow[0,+\infty] measurable then the Lebesgue integral of \varphi on E is\int_Ef(x)\mathrm dx:=\sup_{\varphi(x)\leq f(x),x\in E}\left\{\int_E\varphi(x)\mathrm{d...
In this paper, we begin by introducing some fundamental concepts and results in measure theory and in the Lebesgue theory of integration. We then introduce some functional-analytic concepts and results that will be necessary for the proof of the Lebesgue-Radon-Nikodym Theorem. In the Appendix, we...
We discuss the fundamental convergence theorems of Fatou, B. Levi, and Lebesgue, saying that under certain assumptions, the integral of a limit of a sequence of functions equals the limit of the integrals. Instructive examples show the necessity of those assumptions. As an application, results ...
这本书讲测度论的基本内容,从最基本的东西讲起,对学生的“数学成熟度”要求很低,适合本科生学习。相比而言,很多测度论教材,针对的其实是低年级研究生。我用这本书教了一学期,感觉难度不比周民强的《实变函数》高,但是学到的东西比周民强的更有用。周民强的书,或者说,基本上所有国内的本科《实变函数》教材,都...
It is clear from concavity of Φp that the linear-LSIs (2) are obtained by taking 1αp=ddx|x=0Φp. We briefly review the history of such inequalities: • For a Lebesgue measure on Rn and E(f,g)=∫(∇f,∇g) the p=2 inequality takes the form:(13)∫Rnh2(x)lnh2dx≤...
The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guil...
However, he remarked in 196413 that “the generation times of the organisms which have, at a given time, completed their life span during the previous history of the culture do not compose” h;“they compose the carrier distribution” g. In fact, only information regarding the cell’s age ...
In Probability Theory: Philosophy, Recent History and Relations to Science; Hendricks, V.F., Pederson, S.A., Jorgensen, K.F., Eds.; Kluwer: Amsterdam, the Netherlands, 2001; pp. 167–178. [Google Scholar] Williamson, J. Countable additivity and subjective probability. Br. J. Philos. ...
The inequality (1.7) is sharp, and in the case q=p it is equivalent to the famous Hardy inequality (see [23], [28] for the history of the Hardy inequality). In recent paper, Cassani, Ruf and Tarsi [18] extend the inequality (1.7) to the case p<q≤∞. Our next main result ...