The purpose of this paper is to prove that there exist measures
d
μ
(
x
)
=
γ
(
x
)
d
x
d\mu (x)=\gamma (x)dx
, with
γ
(
x
)
=
γ
0
(
|
x
|
)
\gamma (x)=\gamma _{0}(|x|)
and
γ
0
\gamma _{0}
being a decreasing and positive function, such that the Hardy-Littlewood maximal operator,
M
μ
\mathcal {M}_{\mu }
, associated to the measure
μ
\mu
does not map
L
μ
p
(
R
n
)
L^{p}_{\mu }(\mathbb {R}^{n})
into weak
L
μ
p
(
R
n
)
L^{p}_{\mu }(\mathbb {R}^{n})
, for every
p
>
∞
p>\infty
. This result answers an open question of P. Sjögren and F. Soria.