For certain positive Borel measures
μ
\mu
on
R
{\mathbf {R}}
and for
T
μ
{T_\mu }
any of three naturally associated maximal function operators of Hardy-Littlewood type, the weight pairs
(
u
,
υ
)
(u,\upsilon )
for which
T
μ
{T_\mu }
is of weak type
(
p
,
p
)
,
1
≤
p
>
∞
(p,p),1 \leq p > \infty
, and of strong type
(
p
,
p
)
,
1
>
p
>
∞
(p,p),1 > p > \infty
, are characterized. Only minimal assumptions are placed on
μ
\mu
; in particular,
μ
\mu
need not satisfy a doubling condition nor need it be continuous.