Self-similar measure

Remarks

Each $S_i$ is a contraction. Hutchinson's theorem says that there is exactly one compactly supported such $\mu$. It is the fixed point of a contraction on a space of measures, and its support is the attractor of $(S_i)$. Example: on $\R$ let $S_1(x) = x/2$, $S_2(x) = x/2 + 1/2$ and $p_1 = p_2 = 1/2$, and let $\lambda$ be Lebesgue measure on $[0,1]$. The measure $(S_1)_* \lambda$ is uniform on $[0, 1/2]$ with density $2$, and $(S_2)_* \lambda$ is uniform on $[1/2, 1]$ with density $2$. Their average has density $1$ on $[0,1]$, so it equals $\lambda$, and $\lambda$ is self-similar. Non-example: with the same maps and $p_1 = 1/3$, $p_2 = 2/3$, the average has density $2/3$ on $[0, 1/2]$, so $\lambda$ is not self-similar for these weights. The Cantor measure is self-similar for $S_1(x) = x/3$, $S_2(x) = x/3 + 2/3$ and $p_1 = p_2 = 1/2$.