For a mass distribution $\mu$ defined on the middle-third Cantor set (with left intervals $E_1$ assigned mass $P_1$ and right intervals masses of $P_2$ with the ratios kept this way throughout the construction), show that
$$\mu(A)=\sum_{i=1}^{2} P_{i}(S_{i}^{-1}(A))$$
for Borel A, where $S_1(x)=\frac{x}{3}$ and $S_2(x)=\frac{x}{3}+\frac{2}{3}$.
Show that for continuous $g:\mathbb{R}\to\mathbb{R}$,
$$\int g(x) d\mu (x) = \sum_{i=1}^{2}P_{i}\int g(S_{i} (x)) d\mu (x).$$ I was able to show this is true for single intervals $I_{I_1, I_2,\cdots, I_k}$ in the $k$th stage of construction just by saying one of the transformations yields the empty set each time, but I do not know how to extend this for Borel A.