Let
G
G
be a
σ
\sigma
-compact locally compact nondiscrete group and let
Q
Q
be a
G
G
-invariant ideal of
L
∞
(
G
)
L^{\infty }(G)
. We denote the set of left invariant means
m
m
on
L
∞
(
G
)
L^{\infty }(G)
that are zero on
Q
Q
(i.e.
m
(
f
)
=
0
m(f) = 0
for all
f
∈
Q
f\in Q
) by
L
I
M
Q
LIM_{Q}
. We show that, when
G
G
is amenable as a discrete group and the closed
G
G
-invariant subset of the spectrum of
L
∞
(
G
)
L^{\infty }(G)
corresponding to
Q
Q
is a
G
δ
G_{\delta }
-set,
L
I
M
Q
LIM_{Q}
is very large in the sense that every nonempty
G
δ
G_{\delta }
-subset of
L
I
M
Q
LIM_{Q}
contains a norm discrete copy of
β
N
\beta \mathbb {N}
, where
β
N
\beta \mathbb {N}
is the Stone-
C
ˇ
e
c
h
\mathrm {\check {C}ech}
compactification of the set
N
\mathbb {N}
of positive integers with the discrete topology. In particular, we prove that
L
I
M
Q
LIM_{Q}
has no exposed points in this case and every nonempty
G
δ
G_{\delta }
-subset of the set of left invariant means on
L
∞
(
G
)
L^{\infty }(G)
contains a norm discrete copy of
β
N
\beta \mathbb {N}
.