In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the
σ
-algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the
σ
-algebra generated by the other.
The lemma plays an important role in the conditional expectation in probability theory, where it allows to replace the conditioning on a random variable by conditioning on the
σ
-algebra that is generated by the random variable.
Let
Ω
be a sample space. For a function
f
:
Ω
→
R
n
, the
σ
-algebra generated by
f
is defined as the family of sets
f
−
1
(
S
)
, where
S
are all Borel sets.
Lemma Let
X
,
Y
:
Ω
→
R
n
be random elements and
σ
(
X
)
be the
σ
algebra generated by
X
. Then
Y
is
σ
(
X
)
-measurable if and only if
Y
=
g
(
X
)
for some Borel measurable function
g
:
R
n
→
R
n
.
The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one.
By definition,
Y
being
σ
(
X
)
-measurable is the same as
Y
−
1
(
S
)
∈
σ
(
X
)
for any Borel set
S
, which is the same as
σ
(
Y
)
⊂
σ
(
X
)
. So, the lemma can be rewritten in the following, equivalent form.
Lemma Let
X
,
Y
:
Ω
→
R
n
be random elements and
σ
(
X
)
and
σ
(
Y
)
the
σ
algebras generated by
X
and
Y
, respectively. Then
Y
=
g
(
X
)
for some Borel measurable function
g
:
R
n
→
R
n
if and only if
σ
(
Y
)
⊂
σ
(
X
)
.
