How to calculate expectation of Cantor distribution without using p = 0.5

Question

The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following:

X is characterized by the following conditions: $3X = \epsilon + X'$, $\epsilon = 0$ or 2, and $E(\epsilon) = 1$.