Let
G
G
be a
σ
\sigma
-compact infinite locally compact group, and let
L
I
M
LIM
be the set of left invariant means on
L
∞
(
G
)
{L^\infty }(G)
. We prove in this paper that if
G
G
is amenable as a discrete group, then
L
I
M
LIM
has no exposed points. We also give another proof of the Granirer theorem that the set
L
I
M
(
X
,
G
)
LIM(X,G)
of
G
G
-invariant means on
L
∞
(
X
,
β
,
p
)
{L^\infty }(X,\beta ,p)
has no exposed points, where
G
G
is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space
(
X
,
β
,
p
)
(X,\beta ,p)
.