For a Radon measure
μ
\mu
on
R
,
\mathbb {R},
we show that
L
∞
(
μ
)
L^{\infty }(\mu )
is invariant under the group of translation operators
T
t
(
f
)
(
x
)
=
f
(
x
−
t
)
(
t
∈
R
)
T_t(f)(x) = {f(x-t)}\ (t \in \mathbb {R})
if and only if
μ
\mu
is equivalent to the Lebesgue measure
m
m
. We also give necessary and sufficient conditions for
L
p
(
μ
)
,
1
≤
p
>
∞
,
L^p(\mu ),\ 1 \leq p > \infty ,
to be invariant under the group
{
T
t
}
\{ T_t\}
in terms of the Radon-Nikodým derivative w.r.t.
m
m
.