Measure Theory

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 unfortunate.
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 handout
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, R2, and R3, provided those sets are reasonably
nice, in particular not too irregularly shaped. In R2, 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.