Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 705
The are several kinds of relationships between measures on a given measurable
space or related measurable spaces that are interesting. Some of these
relationships are equivalence relations, but some are not symmetric. The interesting
relationship are generally transitive, however, and so an ordering on
the space of measures could be constructed.
Definition 0.1.23 (dominating measure; absolute continuity; equivalence)
Given measures ν and µ on the same measurable space, (Ω, F), if ∀A ∈ F
ν(A) = 0 ⇒ µ(A) = 0,
then µ is said to be dominated by ν and we denote this by
µ ≪ ν.
In this case we also say that µ is absolutely continuous with respect to ν.
If µ ≪ ν and ν ≪ µ, then µ and ν are equivalent, and we write
µ ≡ ν.
The definition says that µ ≪ ν iff every ν-negligible set is a µ-negligible
set.
If µ is finite (that is, if µ(A) < ∞ ∀A ∈ F), the absolute continuity of µ
with respect to ν can be characterized by an ɛ-δ relationship as used in the
definition of absolute continuity of functions (Definition 0.1.32): Given that
µ is finite, µ is absolutely continuous with respect to ν iff for any A ∈ F and
for any ɛ > 0, there exists a δ such that
ν(A) < δ ⇒ µ(A) < ɛ.
Absolute continuity is a linear relationship; that is, if λ, µ, and ν are
measures on (Ω, F) and a ∈ IR then
(Exercise.)
λ ≪ ν and µ ≪ ν =⇒ (aλ + µ) ≪ ν. (0.1.15)
Definition 0.1.24 (singular measure)
Given measures ν and µ on the same measurable space, (Ω, F), if there exists
two disjoint sets A and B in F such that A ∪ B = Ω and for any measurable
set A1 ⊆ A, ν(A1) = 0, while for any measurable set B1 ⊆ B, µ(B1) = 0 then
the pair of measures ν and µ is said to be singular . We denote this property
as
ν ⊥ µ.
Theory of Statistics c○2000–2013 James E. Gentle