MeasureTheoryLectureNotesTimeless - Economics 204 Lecture...

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Unformatted text preview: Economics 204 Lecture Notes on Measure and Probability Theory This is a slightly updated version of the Lecture Notes used in 204 in the summer of 2002. The measure-theoretic foundations for probability theory are assumed in courses in econometrics and statistics, as well as in some courses in microeconomic theory and finance. These foundations are not developed in the classes that use them, a situation we regard as very unfor- tunate. The experience in the summer of 2002 indicated that it is impossible to develop a good understanding of this material in the brief time available for it in 204. Accordingly, this material will not be covered in 204. This hand- out is being made available in the hope it will be of some help to students as they see measure-theoretic constructions used in other courses. The Riemann Integral (the integral that is treated in freshman calculus) applies to continuous functions. It can be extended a little beyond the class of continuous functions, but not very far. It can be used to define the lengths, areas, and volumes of sets in R , R 2 , and R 3 , provided those sets are rea- sonably nice, in particular not too irregularly shaped. In R 2 , the Riemann Integral defines the area under the graph of a function by dividing the x-axis into a collection of small intervals. On each of these small intervals, two rectangles are erected: one lies entirely inside the area under the graph of the function, while the other rectangle lies entirely outside the graph. The function is Riemann integrable (and its integral equals the area under its graph) if, by making the intervals sufficiently small, it is possible to make the sum of the areas of the outside rectangles arbitrarily close to the sum of the areas of the inside rectangles. Measury theory provides a way to extend our notions of length, area, volume etc. to a much larger class of sets than can be treated using the Riemann Integral. It also provides a way to extend the Riemann Integral to Lebesgue integrable functions, a much larger class of functions than the continuous functions. The fundamental conceptual difference between the Riemann and Lebesgue integrals is the way in which the partitioning is done. As noted above, the Riemann Integral partitions the domain of the function into small intervals. By contrast, the Lebesgue Integral partitions the range of the function into small intervals, then considers the set of points in the domain on which the value of the function falls into one of these intervals. Let f : [0 , 1] → R . 1 Given an interval [ a, b ) ⊆ R , f − 1 ([ a, b )) may be a very messy set. However, as long as we can assign a “length” or “measure” μ ( f − 1 ([ a, b ))) to this set, we know that the contribution of this set to the integral of f should be be- tween aμ ( f − 1 ([ a, b ))) and bμ ( f − 1 ([ a, b ])). By making the partition of the range finer and finer, we can determine the integral of the function....
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MeasureTheoryLectureNotesTimeless - Economics 204 Lecture...

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