In mathematics, a G-measure is a measure
μ
that can be represented as the weak-∗ limit of a sequence of measurable functions
G
=
(
G
n
)
n
=
1
∞
. A classic example is the Riesz product
G
n
(
t
)
=
∏
k
=
1
n
(
1
+
r
cos
(
2
π
m
k
t
)
)
where
−
1
<
r
<
1
,
m
∈
N
. The weak-∗ limit of this product is a measure on the circle
T
, in the sense that for
f
∈
C
(
T
)
:
∫
f
d
μ
=
lim
n
→
∞
∫
f
(
t
)
∏
k
=
1
n
(
1
+
r
cos
(
2
π
m
k
t
)
)
d
t
=
lim
n
→
∞
∫
f
(
t
)
G
n
(
t
)
d
t
where
d
t
represents Haar measure. The
It was Keane who first showed that Riesz products can be regarded as strong mixing invariant measure under the shift operator
S
(
x
)
=
m
x
mod
1
. These were later generalized by Brown and Dooley to Riesz products of the form
∏
k
=
1
∞
(
1
+
r
k
cos
(
2
π
m
1
m
2
⋯
m
k
t
)
)
where
−
1
<
r
k
<
1
,
m
k
∈
N
,
m
k
≥
3
.
