If G is a discrete group and
x
∈
G
x \in G
then
x
∼
x^\sim
denotes the homeomorphism of
β
G
\beta G
onto
β
G
\beta G
induced by left multiplication by x. A subset K of
β
G
\beta G
is said to be invariant if it is closed, nonempty and
x
∼
∅
K
⊂
K
x^\sim \emptyset K \subset K
for each
x
∈
G
x \in G
. Let
M
L
(
G
)
ML(G)
denote the set of left invariant means on G. (They can be considered as measures on
β
G
\beta G
.) Let G be a countably infinite amenable group and let K be an invariant subset of
β
G
\beta G
. Then the nonempty
w
∗
{w^ \ast }
-compact convex set
M
(
G
,
K
)
=
{
ϕ
∈
M
L
(
G
)
:
suppt
ϕ
⊂
K
}
M(G,K) = \{ \phi \in ML(G):{\text {suppt}}\phi \subset K\}
has no exposed points (with respect to
w
∗
{w^ \ast }
-topology). Therefore, it is infinite dimensional.