Chapter 3: Random Variables

3.1. Definitions. -measurable function, , , . 3.2. Elementary Propositions on measurability. 3.3. Lemma. Sums and products of measurable functions are measurable. 3.4. Composition Lemma. 3.5. Lemma on measurability of infs, lim infs of functions. 3.6. Definition. Random variable. 3.7. Example. Coin tossing. 3.8. Definition. -algebra generated by a collection of functions on . 3.9. Definitions. Law, Distribution Function. 3.10. Properties of distribution functions. 3.11. Existence of random variable with given distribution function. 3.12. Skorokod representation of a random variable with prescribed distribution function. 3.13. Generated -algebras - a discussion. 3.14. The Monotone-Class Theorem.

Random variables

A function (or ) is called -measurable if . , and are classes of general, non-negative and bounded measurable functions, respectively. is called Borel if it is -measurable where is a topological space.
  • It is enough to see that either collection of sets , , , or belongs to .
  • If for , then .
  • If for , then , and for , .
A random variable on is an element of .

-algebras generated by functions

Given a collection of functions , the -algebra generated by is the smallest -algebra on such that for each , is -measurable.
  • Intuitively consists of exactly those events for which, for every , whether is determined by the values of .
  • is -measurable iff there is a Borel function such that .
    • is -measurable iff there is a Borel function such that .
    • is -measurable iff there is a countable sequence , , and a Borel function such that .

Law and distribution functions

The law of a random variable is .
The distribution function of is .
  • is a distribution function of some random variable iff is a non-decreasing right-continuous function such that and .
  • The measure associated to such that for all is called the Lebesgue-Stieltjes measure.
  • (Skorokhod representation.) If , then and have distribution function and law .

Monotone-Class Theorem

Let be a vector space over of bounded functions such that , and for in , if , and is bounded, then .
Then, if for some -system , contains all indicator functions of sets of , then .