6-ConceptsProbability Page 172 Wednesday, February 4, 2004 3:00 PM
172 The Mathematics of Financial Modeling and Investment Management■ Property 3. M(∪ Ai) = ∑M(Ai) for every finite or countable collection
of disjoint events {Ai} such that Ai ∈ ℑIf M is a measure defined over a σ-algebra ℑ, the triple (Ω,ℑ,M) is
called a measure space (this term is not used if ℑ is an algebra). Recall
that the pair (Ω,ℑ) is a measurable space if ℑ is a σ-algebra. Measures in
general, and not only probabilities, can be uniquely extended from an
algebra to the generated σ-algebra.RANDOM VARIABLES
Probability is a set function defined over a space of events; random vari-
ables transfer probability from the original space Ω into the space of
real numbers. Given a probability space (Ω,ℑ,P), a random variable X is
a function X(ω) defined over the set Ω that takes values in the set R of
real numbers such that(ω: X(ω) ≤ x) ∈ ℑfor every real number x. In other words, the inverse image of any inter-
val (–∞,x] is an event. It can be demonstrated that the inverse image of
any Borel set is also an event.
A real-valued set function defined over Ω is said to be measurable
with respect to a σ-algebra ℑ if the inverse image of any Borel set
belongs to ℑ. Random variables are real-valued measurable functions. A
random variable that is measurable with respect to a σ-algebra cannot
discriminate between events that are not in that σ-algebra. This is the
primary reason why the abstract and rather difficult concept of measur-
ability is important in probability theory. By restricting the set of events
that can be identified by a random variable, measurability defines the
“coarse graining” of information relative to that variable. A random
variable X is said to generate G if G is the smallest σ-algebra in which it
is measurable.INTEGRALS
In Chapter 4 on calculus we defined the integral of a real-valued function
on the real line. However, the notion of the integral can be generalized to
a general measure space. Though a bit technical, these definitions are
important in the context of probability theory.