Lecture 6: Measure theory and integration. Part A.

In stochastic dynamic programming problems, where stochastic shocks disturb the one-period return function, we try to extend the results we worked out for deterministic problems. Again, the starting point is the Bellman equation, which in one way or another contains the expected value of the value function standing on the left. In some cases this extension is simple: When the stochastic shock takes on only a few number of discrete values, the expected value of the functional equation next period is a simple probability-weighted average. In cases where shocks are continuous, hence they can take on infinitely many values, we need to calculate the expected function in an integral done with respect to a probability measure describing the distribution of the shock. Integration thus plays a key role in stochastic dynamic programming, and to be integrable, the value function is supposed to meet some requirements.

In this lecture we go over the main results of measure theory and integration: How to define measures on measurable spaces; what such measurable spaces are at all; how the features of collections or families of subsets of a space are related to certain characteristics of measures, and how these relations constrain us in defining measures; what characteristics describe the special class of measures that we call 'probability measures'; and how they turn measurable spaces into a special class called 'probability spaces'. As we are going to use integration in value function iteration, the integral we use must be such that makes such iterative manipulations possible. For this reason, we use the more flexible but less known Lebesgue integral, instead of the commonly known Riemann integral. Good news: In most cases, the two integrals lead to the same result, while the Lebesgue integral is more useful when it comes to attacking some esoteric functions.

Chapter 7 of the textbook is very brief and technical, so attendees are encouraged to use George Bartle's 'The elements of integration' (1966). Given the huge amount of formal work we must complete before moving on, Lecture 6 comes in two parts (Part A and B).

Title page:(00:00)
Introduction:(00:11)
Introduction:(08:03)
Agenda for the lecture:(14:19)
Measurable spaces:(18:39)
Measurable spaces:(27:49)
Exercise 7.2:(32:42)
Exercise 7.3:(33:53)
Measures:(35:51)
Exercise 7.4:(42:52)
Exercise 7.5:(49:59)
Measures:(53:20)
Measures:(60:23)