Let
G
G
be an infinite
σ
\sigma
-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on
G
G
. These spaces include the convolution measure algebra
M
(
G
)
M(G)
, the Lebesgue spaces
L
p
(
G
)
{L^p}(G)
, where
1
≤
p
≤
∞
1 \leq p \leq \infty
, and certain combinations thereof. Among other things, it will be shown that
C
(
G
)
C(G)
has many discontinuous translation invariant functionals, provided that
G
G
is amenable. This solves a problem of G. H. Meisters.