A well-known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty }$ where $(f_n)_{n=1}^{\infty }$ is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a

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... ary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t,$ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.