Theory of Statistics - George Mason University

(why?), and so
We also have
λ
0.1 Measure, Integration, and Functional Analysis 713
k
Vk
1 ≤ λ
k
Vk
≤ 3.
=
λ(Vk) =
λ(V ),
which must be either 0 or infinite, in either case contradicting
1 ≤
λ(V ) ≤ 3,
k
k
which follows only from the properties of measures and the assumption that
V is Lebesgue measurable. We therefore conclude that V is not measurable.
0.1.4.9 Borel Measurable Functions
We will now consider real-valued functions; that is, mappings into IR d . The
domains are not necessarily real-valued. We first identify two useful types of
real-valued functions.
Definition 0.1.27 (indicator function)
The indicator function, denoted IS(x) for a given set S, is defined by IS(x) = 1
if x ∈ S and IS(x) = 0 otherwise.
Notice that I −1
S [A] = ∅ if 0 /∈ A and 1 /∈ A; I−1
S [A] = S if 0 /∈ A and 1 ∈ A;
I −1
S [A] = Sc if 0 ∈ A and 1 /∈ A; and I −1
S [A] = Ω if 0 ∈ A and 1 ∈ A.
Hence, σ(IS) is the second most trivial σ-field we referred to earlier; i.e.,
σ(S) = {∅, S, Sc , Ω}.
Definition 0.1.28 (simple function)
If A1, . . ., Ak are measurable subsets of Ω and a1, . . ., ak are constant real
numbers, a function ϕ is a simple function if for ω ∈ Ω,
ϕ(ω) =
where IS(x) is the indicator function.
k
k
aiIAi(ω), (0.1.26)
i=1
Recall the convention for functions that we have adopted: the domain of the
function is the sample space; hence, the subsets corresponding to constant
values of the function form a finite partition of the sample space.
Definition 0.1.29 (Borel measurable function)
A measurable function from (Ω, F) to (IR d , B d ) is said to be Borel measurable
with respect to F.
Theory of Statistics c○2000–2013 James E. Gentle