Based on the seminal work of Hutchinson, we investigate properties of {\em $α$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $α$ is a vector of nonnegative weights summing to $1$, and the corresponding weighted Cantor measure $μ^α$ is the unique Borel probability measure on $[0,1]$ satisfying $ μ^α(E) = \sum_{ n=0 }^{N-1} α_nμ^α( \varphi_n^{-1}(E) )$ where $\varphi_n: x\mapsto (x+n)/N$. In Sections 1 and 2 we examine several general properties of the measure $μ^α$ and the associated Legendre polynomials in $L_{μ^α}^2[0,1]$. In Section 3, we (1) compute the Laplacian and moment generating function of $μ^α$, (2) characterize precisely when the moments $I_m = \int_{[0,1]}x^m\,dμ^α$ exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first $m$ moments within uniform error $\varepsilon$ in $O( (\log\log(1/\varepsilon))\cdot m\log m )$. We also state analogous results in the natural case where $α$ is {\em palindromic} for the measure $ν^α$ attained by shifting $μ^α$ to $[-1/2,1/2]$.