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 MonotoneClass Theorem.
Random variables
A function
(or
) is called
measurable if
.
,
and
are classes of general, nonnegative 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 nondecreasing rightcontinuous function such that and .
 The measure associated to such that for all is called the LebesgueStieltjes measure.
 (Skorokhod representation.) If , then and have distribution function and law .
MonotoneClass 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
.