**Unformatted text preview: **to be able to compute P {X ∈ B } = P {ω : X (ω ) ∈ B } = P {X −1 (B )}
for every Borel set B , then it must be the case that X −1 (B ) is an event (which is to say that
X −1 (B ) ∈ F for every B ∈ B ).
11–2 Deﬁnition. A real-valued function X : Ω → R is said to be a random variable if X −1 (B ) ∈ F
for every Borel set B ∈ B .
Note that when we say let X be a random variable, we really mean let X be a function from
the probability space (Ω, F , P) to the real numbers endowed with the Borel σ -algebra (R, B )
such that X −1 (B ) ∈ F for every B ∈ B . Hence, when we deﬁne a random variable, we
should really also state the underlying probability space as the domain space of X . Since
every random variable we will consider is real-valued, our codomain (or target) space will
always be R endowed with the Borel σ -algebra B . If we want to stress the domain space and
codomain space, we will be explicit and write X : (Ω, F , P) → (R, B ).
Example 11.2. Perhaps the simplest example of a random variable is the indicator function
of an event. Let (Ω, F , P) be a probability space and suppose that A ∈ F is an event. Let
X : Ω → R be given by
1, if ω ∈ A,
X (ω ) = 1A (ω ) =
0, if ω ∈ A.
/
For B ∈ B , we ﬁnd ∅, A,
−1
(1A ) (B ) =
Ac , Ω, if
if
if
if 0 ∈ B ,
0 ∈ B ,
0 ∈ B,
0 ∈ B, 1 ∈ B,
1 ∈ B,
1 ∈ B,
1 ∈ B. Thus, since ∅, A, Ac , and Ω belong to F , we see that for any B ∈ B we necessarily have
X −1 (B ) = (1A )−1 (B ) ∈ F proving that X is a random variable. 11–3...

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