- Source: Equivalence (measure theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.
Definition
Let
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
be two measures on the measurable space
(
X
,
A
)
,
{\displaystyle (X,{\mathcal {A}}),}
and let
N
μ
:=
{
A
∈
A
∣
μ
(
A
)
=
0
}
{\displaystyle {\mathcal {N}}_{\mu }:=\{A\in {\mathcal {A}}\mid \mu (A)=0\}}
and
N
ν
:=
{
A
∈
A
∣
ν
(
A
)
=
0
}
{\displaystyle {\mathcal {N}}_{\nu }:=\{A\in {\mathcal {A}}\mid \nu (A)=0\}}
be the sets of
μ
{\displaystyle \mu }
-null sets and
ν
{\displaystyle \nu }
-null sets, respectively. Then the measure
ν
{\displaystyle \nu }
is said to be absolutely continuous in reference to
μ
{\displaystyle \mu }
if and only if
N
ν
⊇
N
μ
.
{\displaystyle {\mathcal {N}}_{\nu }\supseteq {\mathcal {N}}_{\mu }.}
This is denoted as
ν
≪
μ
.
{\displaystyle \nu \ll \mu .}
The two measures are called equivalent if and only if
μ
≪
ν
{\displaystyle \mu \ll \nu }
and
ν
≪
μ
,
{\displaystyle \nu \ll \mu ,}
which is denoted as
μ
∼
ν
.
{\displaystyle \mu \sim \nu .}
That is, two measures are equivalent if they satisfy
N
μ
=
N
ν
.
{\displaystyle {\mathcal {N}}_{\mu }={\mathcal {N}}_{\nu }.}
Examples
= On the real line
=Define the two measures on the real line as
μ
(
A
)
=
∫
A
1
[
0
,
1
]
(
x
)
d
x
{\displaystyle \mu (A)=\int _{A}\mathbf {1} _{[0,1]}(x)\mathrm {d} x}
ν
(
A
)
=
∫
A
x
2
1
[
0
,
1
]
(
x
)
d
x
{\displaystyle \nu (A)=\int _{A}x^{2}\mathbf {1} _{[0,1]}(x)\mathrm {d} x}
for all Borel sets
A
.
{\displaystyle A.}
Then
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
are equivalent, since all sets outside of
[
0
,
1
]
{\displaystyle [0,1]}
have
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
measure zero, and a set inside
[
0
,
1
]
{\displaystyle [0,1]}
is a
μ
{\displaystyle \mu }
-null set or a
ν
{\displaystyle \nu }
-null set exactly when it is a null set with respect to Lebesgue measure.
= Abstract measure space
=Look at some measurable space
(
X
,
A
)
{\displaystyle (X,{\mathcal {A}})}
and let
μ
{\displaystyle \mu }
be the counting measure, so
μ
(
A
)
=
|
A
|
,
{\displaystyle \mu (A)=|A|,}
where
|
A
|
{\displaystyle |A|}
is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is,
N
μ
=
{
∅
}
.
{\displaystyle {\mathcal {N}}_{\mu }=\{\varnothing \}.}
So by the second definition, any other measure
ν
{\displaystyle \nu }
is equivalent to the counting measure if and only if it also has just the empty set as the only
ν
{\displaystyle \nu }
-null set.
Supporting measures
A measure
μ
{\displaystyle \mu }
is called a supporting measure of a measure
ν
{\displaystyle \nu }
if
μ
{\displaystyle \mu }
is
σ
{\displaystyle \sigma }
-finite and
ν
{\displaystyle \nu }
is equivalent to
μ
.
{\displaystyle \mu .}