Let
G
G
be a locally compact group. If
G
G
is compact, let
L
0
p
(
G
)
L_0^p(G)
denote the functions in
L
p
(
G
)
{L^p}(G)
having zero Haar integral. Let
M
1
(
G
)
{M^1}(G)
denote the probability measures on
G
G
and let
P
1
(
G
)
=
M
1
(
G
)
∩
L
1
(
G
)
{\mathcal {P}^1}(G) = {M^1}(G) \cap {L^1}(G)
. If
S
⊆
M
1
(
G
)
S \subseteq {M^1}(G)
, let
Δ
(
L
p
(
G
)
,
S
)
\Delta ({L^p}(G),S)
denote the subspace of
L
p
(
G
)
{L^p}(G)
generated by functions of the form
f
−
μ
∗
f
f - \mu \ast f
,
f
∈
L
p
(
G
)
f \in {L^p}(G)
,
μ
∈
S
\mu \in S
. If
G
G
is compact,
Δ
(
L
p
(
G
)
,
S
)
⊆
L
0
p
(
G
)
\Delta ({L^p}(G),S) \subseteq L_0^p(G)
. When
G
G
is compact, conditions are given on
S
S
which ensure that for some finite subset
F
F
of
S
S
,
Δ
(
L
p
(
G
)
,
F
)
=
L
0
p
(
G
)
\Delta ({L^p}(G),F) = L_0^p(G)
for all
1
>
p
>
∞
1 > p > \infty
. The finite subset
F
F
will then have the property that every
F
F
-invariant linear functional on
L
p
(
G
)
{L^p}(G)
is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if
1
≤
p
≤
∞
1 \leq p \leq \infty
, conditions are given upon
G
G
, and upon subsets
S
S
of
M
1
(
G
)
{M^1}(G)
whose elements satisfy certain growth conditions, which ensure that
L
p
(
G
)
{L^p}(G)
has discontinuous,
S
S
-invariant linear functionals. The results are applied to show that for
1
≤
p
≤
∞
1 \leq p \leq \infty
,
L
p
(
R
)
{L^p}(\mathbb {R})
has an infinite, independent family of discontinuous translation invariant functionals which are not
P
1
(
R
)
{\mathcal {P}^1}(\mathbb {R})
-invariant.