In Part B of Lecture 6 we complete our analysis on measures, measurable functions, and integration. We start the discussion with two extension theorems (by Carathéodory and Hahn) that make it possible to define a measure to a σ-algebra by extending a measure from an algebra (which is easier to handle), where the extension is unique. The next step is to introduce measurable functions which we need in order to understand the expected values of functions. Note that in any Bellman equation, to say the least, on the right hand side the expected value of the value function always shows up in one way or another. The definition of measurable functions we use, however, is so broad that it covers a vast array of functions including random variables that are interpreted as functions of the state (just think about the case where your action of popping out to the shop for a bar of chocolate or not depends on the outcome of tossing a coin--the state can be head or tail, and your action is a function of this state, depending on how you defined this function). We also study some characteristics of measurable functions: How convergence preserves measurability, and how we can approximate measurable functions with simple functions. We will make good use of these features as they are all instrumental in ensuring some key convergence properties of the Lebesgue integral. Of these properties, the ones established by the Monotone Convergence Theorem and the related Lebesgue Dominated Convergence Theorem are the most important to us. According to the former, the integral of a non-negative function f is the limit of a series of integrals of functions with respect to the same measure if the series of functions converges to f pointwise. According to the latter, we have a similar result for functions that may take either sign, supposing a series of integrable functions converges almost everywhere (that means the state space can be divided into two subsets: There is convergence on the one, while the other partition measures zero) to f, and if there is an integrable function g that is greater than or equal to f in modulus.

Title page:(00:00)
Measures - The Carathéodory and Hahn extension theorems:(00:10)
Exercise 7.7:(05:23)
Measurable functions:(07:36)
Pointwise convergence preserves measurability:(14:38)
Approximation of measurable functions by simple functions:(15:29)
Exercise 7.10:(19:18)
Integration:(23:40)
The Monotone Convergence Theorem:(32:40)
Integration and convergence almost everywhere:(38:49)
Integral of functions taking either sign:(41:38)
The Lebesgue dominated convergence theorem:(43:27)
Product spaces:(45:28)