08-01. Conditional expectation - Radon-Nikodym theorem and conditional expectation.

This video defines the concept of conditional expectation (of a random variable given a sigma-algebra) and shows its existence and uniqueness using the Radon-Nikodym theorem in measure theory. This is Sections 2.1 and 2.3 of my Stochastic Modeling book.

[irelandrone]

Can you give an example of such $A_n$ where $A_i\ cap A_j = \phi$ but $A_n$ forms a nondecreasing sequence so that I can apply MCT? I did not understand the sigma additivity.

[theodorevassilakis]

In the proof of the existence of the conditional random variable Z around time stamp 28:30 isn’t it necessary to justify what happens if the the integral if X^+ and X^- are infinite? Presumably that’s covered by vthe L^1 assumption on X but it would be helpful to say how.

[theodorevassilakis]

In the video the sub sigma algebra G is characterized as “less” information than the full sigma algebra F. If F represents the full set of possible events in the random trial and G represents a restriction of what can actually happen, isn’t it also reasonable to view G as providing less uncertainty and thus more information about the outcome?

[irelandrone]

Why do you need positive measurable, Dear Professor? by positive do you mean $\phi >0 \forall x\in \Omega$