Real Analysis with Economic Applications - Content Preface pps

... Set Theory (Chapters A-B) II. Analysis on Metric Spaces (Chapters C-E) III. Analysis on Linear Spa ces (Chapters F-H) IV. Analysis on Metric/Normed Linear Spaces (Chapters I-K) Part I pro v id ... books on real analysis pertains to the choice of topics. I put much more emphasis on topics that are immediately relevant for economic theory, and omit some standard t hem es of real...

Ngày tải lên: 04/07/2014, 10:20

... all bounded real sequences, th at is,  ∞ := {(x m ) ∈ R ∞ :sup{|x m | : m ∈ N} < ∞} . It is implicitly understood that this set i s e ndowed with the metric d ∞ :  ∞ × ∞ → R + with d ∞ ((x m ) ... R ∞ since the mem bers of such a space are real sequences that are either bounde d or that satisfy some form of a summability condition (that ensures that d p is real- valued). Indeed...

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... semicon tinuity of a function like ϕ • whic h ma y be extended real- valued (ev en w hen ϕ is real- valued). 20 Exercise 32. Let ϕ be any real function defined on a metric space X. Pro ve: (a) ϕ • ≤ ... prove this approximation-by-continuous-functions theorem, but you might want to try it out for yourself in the case where X is a compact interval. 176 3.2 The Local-to-Global Method Since a...

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... corollary. The Bolzano-Weierstrass Theorem. 34 Ev ery bounded real sequence has a con- v ergen t subsequence. Exercise 37. Show that every unbounded real sequence has a subsequence that di- v erges to ... e difference-quotient map Q f,x : I\{x} → R by Q f,x (t):= f(t)−f(x) t−x . If the right-limit of this map at x exists as a real number, that is, Q f,x (x+) ∈ R, then f is said to be rig...

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... to be a well-ordering if ev ery nonempty subset of X has a -minimum. In this case X is said to be w e ll-ordered by ,and(X, ) is called a well-ord e red s et,orshortlyawoset. Well-ordered sets ... Principle, it contains a ⊇-maximal loset, say (Z i , ). By -densene ss of Y , Z i has neither a -maxim um nor a -minimum. Moreove r, by its maximalit y, it i s -d en s e in itself. By Cor .....

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... open subset O of R n + with B(p, ι) ∩ O = ∅. To derive a con trad iction, suppose that for every m ∈ N (howev er large), there exists an (n +1)-vector (p m , ι m ) within 1 m -neighborh ood of (p, ... R n .IfΓ is a convex-valued self-co rres pondenc e on X that has a closed graph, then Γ has a fix ed point, tha t is, there exists an x ∈ X with x ∈ Γ(x). 20 Sadly, Shizuo Kakutani (191 1-2...

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... uniqueness claims made in t he Anscombe-Aumann Ex- pected Utility Theorem. ∗ Exercise 46. Obtain the counterpart of the Anscombe-Aumann Expected Utility Theorem (with modifying some of the axioms ... the risk p refer- ences of the agent is state-independent. 305 for all A, B ⊆ N with A ∩ B = ∅. The capacit y v is said to be additive if v(A ∪ B)=v(A)+v(B) for all A, B ⊆ N with A ∩ B = ∅....

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... al-int X (S), then B\{y, z } ⊆ al-int X (S). (Why?) But this wou ld imply that x ∈ al-int X (S), a contradiction. Th us al-int X (S)∩B = ∅. In turn, this means that al- int X (A) = ∅ an d al-int X (A) ... S (in X), and is denoted by al-cl X (S). (Note. al-c l X (∅)=∅.)Ifal-cl X (S) ⊆ S, we say that S is algebr a ically closed in X. F in ally, the set al-cl X (S)\al-int X (S) is called the al...

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... nscombe-Aumann setup, the prefer- enc e relation  of an ind iv idu al is defined on H Ω,X . If this preference relation is com- plete and it satisfies the Independence, Cont inuity, No-Tr iv iality, ... conclude the section with the proof o f the Gilboa-Schm eidler Theorem. T his is a “deep” result, but on e which is not all that hard to establish once o ne realizes that convex analysis li...

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... Prove: (a) cl X (S)=al-cl(S); (b) bd X S = al-bd X (S); (c) cl X (S)=cl X (al-in t X (S)) = cl X (int X (S)). 444 to check that L is continuou s at 0. So, take any (x m ) ∈ X ∞ with x m → 0. We wish to ... more g en er ally, a convex set with a nonempty interior) also as “large.” E{dpsoh 8. We hav e seen earlier that al-int  p ( p + )=∅ for a ny 1 ≤ p<∞ (Exer- cise G.23). In view of...

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