This talk addresses the principal problem of characterizing non-negative functions w for which Hardy-Littlewood maximal operators associated with Hausdorff contents satisfy weighted Lp-norm inequalities with 1<p<infty and weighted weak L1-norm inequalities. To this end, we introduce a new class of capacitary Muckenhoupt weights, denoted by A_{p,\delta}, depending on the dimension \delta of Hausdorff contents. By proposing a new approach, we then establish Muckenhoupt's theorem, reverse Holder's inequality, self-improving property, and Jones' factorization theorem within this capacitary Muckenhoupt weight framework. Moreover, we reveal the deep connections between A_{p,\delta} with BMO/BLO spaces with respect to Hausdorff content. These results naturally extend the classical theory beyond measure-theoretic frameworks, while also provide innovative proofs---even when reduced to the classical Muckenhoupt A_p weight case (i.e., \delta=n)---that avoid the use of Fubini's theorem, the countable additivity of measures and linearity of the integral.