  For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type
measures on $\mathbb{R}$ supported on sets of Hausdorff dimension
$\alpha<\alpha'$ for which the associated maximal operator is bounded from
$L^p_\delta (\mathbb{R})$ to $L^p(\mathbb{R})$. Maximal theorems for fractal
measures on the line were previously obtained by Laba and Pramanik. The result
here is weaker in that we are not able to obtain $L^p$ estimates; on the other
hand, our approach allows Cantor measures that are self-similar, have
arbitrarily low dimension $\alpha>0$, and have no Fourier decay. The proof is
based on a decoupling inequality similar to that of Laba and Wang.
