CONTENTS Chapter 5 focuses primarily on Lebesgue measure and integration since almost all applications that economists study are covered by this case and because it is easy to conceptualize the notion of distance through that of the restriction of an outer measure. We show that the collection of lebesgue measurable sets is a o-algebra and that the collection of borel sets is a subset of the Lebesgue measurable sets. Then we provide a set of convergence theorems for the existence of a Lebesgue integral which are applicable under a wide variety of conditions. Next we introduce general and signed measures where we show that a signed measure can be represented simply by an integral the Radon-Nikodyn Theorem). To prepare for the following chapter, we end by studying a simple function space(the space of integrable functions) and prove it is complete. We study properties such as completeness and compactness in two impor- tant function spaces in Chapter 6: the space of bounded continuous functions (denoted C(X)) and the space of p-integrable functions(denoted Lp(X)).A fundamental result on approximating continuous functions in C(X) is given in a very general set of Theorems by Stone and Weierstrass. Also, the Brouwer Fixed Point Theorem of Chapter 4 on finite dimensional spaces is generalized to infinite dimensional spaces in the Schauder Fixed Point Theorem. Mov ing onto the Lp(X) space, we show that it is complete in the Riesz-Fischer Theorem. Then we introduce linear operators and functionals, as well as the notion of a dual space. We show that one can construct bounded linear functionals on a given set X in the Hahn-Banach Theorem, which is used to prove certain separation results such as the fact that two disjoint convex sets an be separated by a linear functional. Such results are used extensively in economics; for instance, it is employed to establish the Second Welfare Theorem. The chapter ends with nonlinear operators and focuses particu- larly on optimization in infinite dimensional spaces. First we introduce the weak topology on a normed vector space and develop a variational method of optimizing nonlinear functions. Then we consider another method of finding the optimum of a nonlinear functional by dynamic programming Chapter 7 provides a brief overview of general topological spaces and the idea of a homeomorphism (i.e. when two topological spaces X and Y have"similar topological structure"which occurs when there is a one-to-one and onto mapping f from elements in X to elements in Y such that both f and its inverse are continuous). We then compare and contrast topological and metric properties, as well as touch upon the metrizability problem (i finding conditions on a topological space X which guarantee that there existsCONTENTS 11 Chapter 5 focuses primarily on Lebesgue measure and integration since almost all applications that economists study are covered by this case and because it is easy to conceptualize the notion of distance through that of the restriction of an outer measure. We show that the collection of Lebesgue measurable sets is a σ-algebra and that the collection of Borel sets is a subset of the Lebesgue measurable sets. Then we provide a set of convergence theorems for the existence of a Lebesgue integral which are applicable under a wide variety of conditions. Next we introduce general and signed measures, where we show that a signed measure can be represented simply by an integral (the Radon-Nikodyn Theorem). To prepare for the following chapter, we end by studying a simple function space (the space of integrable functions) and prove it is complete. We study properties such as completeness and compactness in two impor￾tant function spaces in Chapter 6: the space of bounded continuous functions (denoted C(X)) and the space of p-integrable functions (denoted Lp(X)). A fundamental result on approximating continuous functions in C(X) is given in a very general set of Theorems by Stone and Weierstrass. Also, the Brouwer Fixed Point Theorem of Chapter 4 on finite dimensional spaces is generalized to infinite dimensional spaces in the Schauder Fixed Point Theorem. Mov￾ing onto the Lp(X) space, we show that it is complete in the Riesz-Fischer Theorem. Then we introduce linear operators and functionals, as well as the notion of a dual space. We show that one can construct bounded linear functionals on a given set X in the Hahn-Banach Theorem, which is used to prove certain separation results such as the fact that two disjoint convex sets can be separated by a linear functional. Such results are used extensively in economics; for instance, it is employed to establish the Second Welfare Theorem. The chapter ends with nonlinear operators and focuses particu￾larly on optimization in infinite dimensional spaces. First we introduce the weak topology on a normed vector space and develop a variational method of optimizing nonlinear functions. Then we consider another method of finding the optimum of a nonlinear functional by dynamic programming. Chapter 7 provides a brief overview of general topological spaces and the idea of a homeomorphism (i.e. when two topological spaces X and Y have ìsimilar topological structureî which occurs when there is a one-to-one and onto mapping f from elements in X to elements in Y such that both f and its inverse are continuous). We then compare and contrast topological and metric properties, as well as touch upon the metrizability problem (i.e. finding conditions on a topological space X which guarantee that there exists