For 2 ≤ p < ∞, α ′ > 2/p, and δ > 0, we construct Cantor-type measures on R supported on sets of Hausdorff dimension α < α ′ for which the associated maximal operator is bounded from L p δ (R) to L p (R). Maximal theorems for fractal measures on the line were previously obtained by Łaba and Pramanik [17]. 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 α > 0, and have no Fourier decay. The proof is based on a decoupling inequality similar to that of Łaba and Wang [18].