n the chapter (
Conditional
probability chapter
) we introduced notions of random events and
illustrated how statements about realization of random variables could be
treated as operations on sets and measured via probability function defined on
these sets. In the present chapter we expand such approach.
The references for this chapter are
[Royden]
,
[Chung]
and
[Kolmogorov]
.
The triple
is called "probability space". Here the
is an event space,
is a
algebra
of subsets of
(see the section
(
Operations on sets and
logical statements
)) and
is a probability measure (p.m.)
,
and
is
additive.
We denote
.
is the Borel field with
and
.
"Borel field
"
is a minimal
algebra
containing all open sets.
Definition
(Borel measurable function) The mapping
is a "Borel measurable function"
iff
The random variable
on
induces the triple
according to the
rule
For a measure
we introduce a (cumulative) distribution function (d.f.)
,
