Hemicontinuous Correspondences
Many problems in economics result in set-valued mappings or correspondences as defined in Section 2.3. For instance, if preferences are linear, a household's demand for goods may described by a correspondence and in game theory we consider best response correspondences. Before defining hemicontinuity, we amend Definition 48 of a correspondence in Section 2.3 in terms of general metric spaces. Definition 269 A correspondence r from a metric space X, dx into a metric space Y, dY is a rule that...
Theorem 116 Nested Intervals Property in R If n n G N is a set
of non-empty, closed, nested intervals in R, then 3x G R such that n 1 n 0. n Proof. Let n an,bn with an lt bn. Since 1 D n, then b1 gt bn gt an. Hence an n G N is bounded above and let a be its sup. To establish the claim, it is sufficient to show a lt bn, Vn G N. Suppose not. Then 3m G N 3bm lt a. Since a sup an n G N , 3ap gt bm. Let q max p,m . Then bq lt bm lt ap lt aq. But bq lt aq contradicts q is a non-empty interval. Thus an lt a lt bn or a G n, Vn G N. If n is not closed, then the...
Optimization of Nonlinear Operators
In this chapter we have dealt with linear operators and functionals. While we showed very deep results in linear functonal analysis - the Riesz Representation Theorem and the Hahn Banach Theorem to name just a few -there are many problems in economics that involve nonlinear operators. For instance, the operator in most dynamic programing problems, such as the growth example suggested in the introduction to this chapter, does not satisfy the linearity property of an operator. In particular, an...
Lebesgue Measure
Before embarking on the general definition of a measure space, in the context of a simple set X R we will introduce the notion of length again just a real-valued function defined on a subset of R , describe desireable properties of a measure space, and describe a simple measure related to length. Definition 326 A set function associates an extended real number to each set in some collection of sets. In R, the length I of an interval I C R is the difference of the endpoints of 1.2 Thus, in the...
Theorem of the Maximum
In economics, often we wish to solve optimization problems where households maximize their utility subject to constraints on their purchases of goods or firms maximize their profits subject to constraints given by their technology. In particular, consider the following example. Example 294 A household has preferences over two consumption goods c1, c2 characterized by a utility function U R R given by U c1, c2 c1 c2. The household has a positive endowment of good 2 denoted u R . The household...
Completion of a metric space
Every metric space can be made compete. The idea is a simple one. Let X, d be a metric space that is not complete. Let CS X be the set of all Cauchy sequences on the incomplete metric space and let lt an gt , lt bn gt CS X . Define as in Definition 26 the equivalence relation by lt xn gt lt yn gt iff limn d xn,yn 0. This relation forms a partition of CS X where in every equivalence class there are all sequences which have the same limit. Let X be the set of all equivalence classes of CS X ....
Exercise 241 Finish the proof of Theorem 52
Definition 53 If H C B, then the inverse image of H under f, denoted f-1 H , is the subset a f a e H C D f . See Figure 2.4.3b. It is important to note that the inverse image is different from the inverse function to be discussed shortly . The inverse function need not exist when the inverse image does. See Example 65. Theorem 54 Let G,H C B. a If G C H,, then f-1 G C f-1 H . b f -1 G D H f-1 G D f -1 H , c f G U H f-1 G U f -1 H , d f -1 G H f -1 G f -1 H . Proof. a If a E f-1 G , then f a E G...
Intermediate value theorem
Theorem 254 Preservation of Connectedness The image of a connected space under a continuous function is connected. Proof. Let f X Y be a continuous function on X and let X be connected. We wish to prove that Z f X is connected. Assume the contrary. Then there exists open disjoint sets A and B such that Z A H Z U B H Z and A H Z , B H Z is a separation of Z into two disjoint, non-empty sets in Z. Then f-1 A H Z f-1 A H f-1 Z f-1 A H X f-1 A and f-1 B H Z f-1 B are disjoint sets whose union is X...
Borel Sets
Since the intersection of a countable collection of open sets need not be open e.g. Example 107 , the collection of all open sets in R is not a a-algebra. By Theorem 87, however, there exists a smallest a-algebra containing all open sets. Definition 123 The smallest a-algebra generated by the collection of all open sets in R, denoted B, is called the Borel a-algebra in R. Just as Example 83 showed in the case of algebras, even though B is the smallest a-algebra containing all open sets, it is...
End of Chapter Problems 1
1. Let D be non-empty and let f D R have bounded range. If Do is a non-empty subset of D, prove that inf f x x D lt inf f x x D0 lt sup f x x D0 lt sup f x x D 2. Let X and Y be non-empty sets and let f X X Y R have bounded range in R. Let f 1 x sup f x,y y Y , f2 y sup f x,y x X Establish the Principle of Iterated Suprema sup f x, y x X,y Y sup f1 x x X sup f2 y y Y We sometimes express this as supx y f x,y supx supy f x,y 3. Let f and f1be as in the preceding exercise and let sup g2 y y Y lt...