Subadditivity of matrix entropy and concentration of. Measure theory chapter 3 from g14fta further topics in analysis 201112 suitable for students with some knowledge of metric and topological spaces. But avoid asking for help, clarification, or responding to other answers. The lebesgue outer measure has a very nice property known as countable subadditivity. In this paper we offer seven proofs of the subadditivity of es, some. Natural risk statistics include var as a special case and therefore provide a theoretical basis for using var along with scenario analysis as a robust risk measure for the purpose of external, regulatory risk measurement. If we represent the distribution by the appropriate random variable x, and let h represent the risk measure functional, then h. A measure theory tutorial measure theory for dummies maya r. Subadditivity is the key property which distinguishes the popular risk measures valueatrisk and expected shortfall es. A test for costs subadditivity in the fishery sector. This is a preliminary version of the book an introduction to measure theory published by the american mathematical society ams. In measure theory we look carefully at various ways to measure the size of a set.
Seven proofs for the subadditivity of expected shortfall. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Subadditivity of matrix entropy and concentration of random. Eric hansen april 20, 2006 1 subadditivity and feketes theorem lemma 1 fekete if fang is subadditive then lim n. Entropy in classical and quantum information theory. Gilbargtrudinger 5, geometric measure theory, geometric group theory, ergodic theory and dynamical systems, and di erential topology and geometry. Given a measure on a semiring j, is it possible to extend it. Probability spaces, distributions, random variables, measure theory scott she eld mit 18. Toward categorical risk measure theory 391 structing a monetary value measure satisfying the axiom from any given monetary value measure. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. In the previous section we dealt however only with semirings.
Among all of the properties of entropy, the most famous one is the ssa, and it is very di. In this sense, a measure is a generalization of the concepts of length, area, volume, etc. Borel measures play a preeminent role in measure theory on rn. I hope that it will be found useful by undergraduates meeting lebesgue measure for the. Classical potential theory attended by an excellent class of graduate students of the department of mathematics of washington university.
Lebesguemeasure on university of california, davis. Toward categorical risk measure theory 393 we sometimes adopt the practice of using a monetary value measure instead of using a monetary risk measure. That was my rst time to teach such a course and, i have to say, besides sporadic knowledge of a few facts directly related to complex analysis, i had no serious knowledge of the subject. In this paper we o er seven proofs of the subadditivity of es, some found in the literature and some not. In this introductory chapter we set forth some basic concepts of measure theory, which will open for abstract lebesgue integration. Volume 2 aims to lay out some of the fundamental results of pure measure theory the radonnikody. Therefore it is natural to ask the following question 1. For a smooth action of zd preserving a borel probability measure, we show that entropy satis. Preliminary measure theory we also conclude the additivity s \ n. It implies that production from only one firm is socially less expensive in terms of average costs than production of a fraction of the original quantity by an equal. Moreover, as formalized by kolmogorov 1933, measure theory provides the foundation of probability. An introduction to measure theory terence tao terry tao.
What we refer to as a measure, many texts will refer to as an outer measure. Probability theory deals with random events and their probabilities. Data, coherent risk measures, and insurance risk measures c. Using the subadditivity of measures of proposition 2. Coherent risk measure, copula, distortion risk measure, extreme value theory, gluevar, tail subadditivity, tail distortion risk measure 1. This preliminary version is made available with the permission of the ams and may not be changed, edited, or reposted at any other website without. A measure on a topological space for which the measurable sets is the borel algebra bx is called a borel measure. Second, it is also a measure of the classical information content per letter, where we nd the maximum amount of infor. A risk measure is a functional mapping a loss or pro. Peng columbia university june 26, 2006 abstract two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivity for random variables, and the insurance risk measure, which assumes additivity for comonotonic random.
Uncertainty theory based on normality, monotonicity, and countable subadditivity was founded in 2007 by liu 2007 and refined in 2010 by liu 2010a. They include functional analysis and partial di erential equations see e. Subadditivity of chance measure journal of uncertainty. R the risk measure is assumed in some way to encapsulate the risk associated with a.
Measuring things already the ancient greeks developed a theory of how to measure length, area, and volume and area of 1. Analogously, a set in a measure space is said to have a. There are many other textbooks on measure theory that cover most or. In this section, some basic concepts about uncertainty theory such as uncertain measure, uncertain variable, uncertainty distribution, and uncertain reliability are given. Risk is the probability that a disaster will happen. But some other properties is not so obvious, such as the strong subadditivity theorem ssa. For instance, subjects in one experiment judged the probability of death from cancer in the united states was 18%, the probability from heart attack was 22%, and the probability of death from other natural causes was 33%. The most important of these conditions is subadditivity, which states that the risk of a portfolio is always less than or equal to the sum of the risks of its parts. We could not do this in a good way for all sets, so we constructed exterior lebesgue measure on the measure space rd,prd, and then restricted to the class of lebesgue measurable. The subadditivity effect is the tendency to judge probability of the whole to be less than the probabilities of the parts example. Recently, quantum information theory attracts more and more attentions for its misterious properties and its potential applications in science and technology 7,8.
Thus, the risk measure of the sum of two assets is bounded above b y the sum of their individual 2 risks. Chance theory is a mathematical methodology for dealing with indeterminacy phenomena involving uncertainty and randomness. For both areas of application the theory is revised in detail and examples are given to show how to apply the concept in practice. We can say that risk is a failure, an unexpected result. Subadditivity is an essential property of some particular cost functions. A measure theory tutorial measure theory for dummies. Seven proofs for the subadditivity of expected shortfall paul embrechts and ruodu wangy october 7, 2015 abstract subadditivity is the key property which distinguishes the popular risk measures valueatrisk and expected shortfall es. The survey lie75 contains a discussion of subadditivity in statistical mechanics, and the monograph rs describes examples in information theory.
In this paper, some properties of chance space are investigated. New class of distortion risk measures and their tail. Finally, let us prove the finite additivity of the measure 1. M mbe an action by measurepreserving, measurable transformations. Volume 1 is intended for those with no prior knowledge of measure theory, but competent in the elementary techniques of real analysis. Thanks for contributing an answer to mathematics stack exchange. An introduction to risk measures for actuarial applications. In the theory of measures, the assumption of additivity is replaced by a. As stated in the following definition, an outer measure is a monotone, countably subadditive, nonnegative, extended realvalued function defined on all subsets. Algebra of sets 5 2 lebesgue measure on r 21 3 measurable functions 33 4 convergence a.
In this work, we focus on the role of subadditivity of entropy in probability. Lecture notes on measure theory and functional analysis. A is closed if and only if for any sequence xn in a such that xn x, x must belong to a only. An overview of financial risk management what is risk. Second, it is also a measure of the classical information content per. First, it is a measure of the quantum information content of letters in the ensemble, speci cally, how many quibits are needed to encode the message without loss of information. This preliminary version is made available with the permission of the ams and may not be changed, edited, or reposted at any other website without explicit written. Countable subadditivity of the lebesgue outer measure. Chapter 2 basic measure theory 5 topology in the euclidean space open set, closed set, compact set properties. Based on this, the subadditivity theorem, nulladditivity theorem, and asymptotic theorem of chance measure are proved. Measure theory measure theory is a rich subject in and of itself. Measures are important not only because of their intrinsic geometrical and probabilistic signi. The theory makes rigorous the notions of length, area and volume, and generalises these notions. Introduction to geometric measure theory stanford university.
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