In descriptive set theory, a tree over a product set
Y
×
Z
is said to be homogeneous if there is a system of measures
⟨
μ
s
∣
s
∈
<
ω
Y
⟩
such that the following conditions hold:
μ
s
is a countably-additive measure on
{
t
∣
⟨
s
,
t
⟩
∈
T
}
.
The measures are in some sense compatible under restriction of sequences: if
s
1
⊆
s
2
, then
μ
s
1
(
X
)
=
1
⟺
μ
s
2
(
{
t
∣
t
↾
l
h
(
s
1
)
∈
X
}
)
=
1
.
If
x
is in the projection of
T
, the ultrapower by
⟨
μ
x
↾
n
∣
n
∈
ω
⟩
is wellfounded.
An equivalent definition is produced when the final condition is replaced with the following:
There are
⟨
μ
s
∣
s
∈
ω
Y
⟩
such that if
x
is in the projection of
[
T
]
and
∀
n
∈
ω
μ
x
↾
n
(
X
n
)
=
1
, then there is
f
∈
ω
Z
such that
∀
n
∈
ω
f
↾
n
∈
X
n
. This condition can be thought of as a sort of countable completeness condition on the system of measures.
T
is said to be
κ
-homogeneous if each
μ
s
is
κ
-complete.
Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.
