Let G be a nondiscrete LCA group,
M
(
G
)
M(G)
the measure algebra of G, and
M
0
(
G
)
{M_0}(G)
the closed ideal of those measures in
M
(
G
)
M(G)
whose Fourier transforms vanish at infinity. Let
Δ
G
,
Σ
G
{\Delta _G},{\Sigma _G}
and
Δ
0
{\Delta _0}
be the spectrum of
M
(
G
)
M(G)
, the set of all symmetric elements of
Δ
G
{\Delta _G}
, and the spectrum of
M
0
(
G
)
{M_0}(G)
, respectively. In this paper this is shown: Let
Φ
\Phi
be a separable subset of
M
(
G
)
M(G)
. Then there exist a probability measure
τ
\tau
in
M
0
(
G
)
{M_0}(G)
and a compact subset X of
Δ
0
∖
Σ
G
{\Delta _0}\backslash {\Sigma _G}
such that for each
|
c
|
⩽
1
|c| \leqslant 1
and each
\[
ν
∈
Φ
Card
{
f
∈
X
:
τ
^
(
f
)
=
c
and
|
ν
^
(
f
)
|
=
r
(
ν
)
}
⩾
2
c
.
\nu \in \Phi \;{\text {Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text {and}}\;|\hat \nu (f)| = r(\nu )\} \geqslant {2^{\text {c}}}.
\]
Here
r
(
ν
)
=
sup
{
|
ν
^
(
f
)
|
:
f
∈
Δ
G
∖
G
^
}
r(\nu ) = \sup \{ |\hat \nu (f)|:f \in {\Delta _G}\backslash \hat G\}
. As immediate consequences of this result, we have (a) every boundary for
M
0
(
G
)
{M_0}(G)
is a boundary for
M
(
G
)
M(G)
(a result due to Brown and Moran), (b)
Δ
G
∖
Σ
G
{\Delta _G}\backslash {\Sigma _G}
is dense in
Δ
G
∖
G
^
{\Delta _G}\backslash \hat G
, (c) the set of all peak points for
M
(
G
)
M(G)
is
G
^
\hat G
if G is
σ
\sigma
-compact and is empty otherwise, and (d) for each
μ
∈
M
(
G
)
\mu \in M(G)
the set
μ
^
(
Δ
0
∖
Σ
G
)
\hat \mu ({\Delta _0}\backslash {\Sigma _G})
contains the topological boundary of
μ
^
(
Δ
G
∖
G
^
)
\hat \mu ({\Delta _G}\backslash \hat G)
in the complex plane.