"Probability" measures on Cantor set I'm not fully acquainted with measure theory, so a detailed explanation may be needed here. 
From what I already understand, the Lebesgue measure on Cantor set (denote it by: CL-measure) $C$ gives probability $\frac{1}{2}$ to each of the the digits $0,2$ for each $x \in C$ expansion in ternary base.
Now, is there any difference in terms of measure of subsets of $C$ when the probabilities are $p,1-p$? To further focus my question, how is the  power law  (namely $c_1r^\alpha \leq \mu(B(x,r)) \leq c_2r^\alpha$ for all $ x \in \operatorname{supp}\mu$) affected by this change? I have seen some remarks about it holding for CL-measure when $\alpha = \frac{\log2}{\log3}$. 
I have found very sparse information about this, and I'll be happy if someone knows a good source to look at.  
 A: The answer to your first question, "is there any difference in terms of measure of subsets of $C$ when the probabilities are $p$, $1-p$?" is: Yes, in fact there's about as much difference as one could possibly imagine.  As $p$ varies, these measures concentrate on pairwise disjoint subsets of $C$.
In more detail: Let $\mu_p$ be the measure on the Cantor set that you obtain by giving the digits $0$ and $2$ the probabilities $p$ and $1-p$, respectively.  (More precisely, this defines a measure on $\{0,2\}$; use it to define the product measure on the set of all infinite sequences of $0$'s and $2$'s; and finally identify those sequences with points of the Cantor set $C$ via base-$3$ expansions.)  Let $A_p$ be the set of those $x\in C$ whose ternary expansion has asymptotically a fraction $p$ of $0$'s; that is, if we let $Z_x(n)$ be the number of zeros among the first $n$ digits in the ternary expansion of $x$, then $x\in A_p$ iff $\lim_{n\to\infty}(Z_x(n)/n)=p$.  Notice that the sets $A_p$ for different values of $p$ are pairwise disjoint.  Furthermore, the strong law of large numbers implies that $\mu_p(A_p)=1$.  So the measures $\mu_p$, for different values of $p$, concentrate on disjoint sets.
A: 
To further focus my question, how is the power law (...) affected by this change?

Yes it is.
To show why, let us first recall the heuristics leading to a power law for the Cantor set $C$ with respect to the Lebesgue measure, that is, when $p=\frac12$. For $x$ in $C$ and $n\geqslant1$, the elements of $C$ in $B(x,3^{-n})$ are roughly those whose $n$ first digits of their ternary expansion coincide with those in the expansion of $x$. This happens with probability $2^{-n}$ hence $\mu(B(x,3^{-n}))\approx2^{-n}$. And indeed, this approximate equivalent reads $\mu(B(x,r))\approx r^\alpha$ with $\alpha=\log2/\log3$.
When the probabilities of the digits $0$ and $2$ are $p$ and $1-p$, the first part of the reasoning subsists, that is, the elements of $C$ in $B(x,3^{-n})$ are roughly those whose $n$ first digits of their ternary expansion coincide with those in the expansion of $x$. But now this happens with probability $p^{N_n(x)}(1-p)^{n-N_n(x)}$, where $N_n(x)$ denotes the number of $0$ in the $n$ first digits of the ternary expansion of $x$, and, for every $p\ne\frac12$, this probability depends on $N_n(x)$. Some consequences follow:


*

*The Hausdorff exponent $\alpha$ of $C$ at $x$, defined by the property that $\mu_p(B(x,r))\approx r^\alpha$ when $r\to0$, may not exist, and indeed it does not exist at those points $x$ such that $N_n(x)/n$ does not converge.

*Every exponent $\alpha$ between $-\log p/\log 3$ and $-\log(1-p)/\log3$ corresponds to an infinite set of points $x$, those such that $N_n(x)/n$ converges to $\nu(\alpha)$ where, for every $a$ in this interval, $\nu(a)$ is defined by the identity
$$
p^{\nu(a)}(1-p)^{1-\nu(a)}3^a=1.
$$

*Almost surely with respect to $\mu_p$, $\alpha=\alpha_p$, where 
$$
\alpha_p=-(p\log p+(1-p)\log(1-p))/\log3,
$$ 
hence $\alpha_p$ can also be defined by the identity $\nu(\alpha_p)=p$.


To sum up, with respect to $\mu_p$, the Hausdorff exponent of $C$ is $\alpha_p$ almost everywhere while the subsets of $C$ where the Hausdorff exponent has any fixed value between $-\log p/\log 3$ and $-\log(1-p)/\log3$ and the subset of $C$  where the Hausdorff exponent does not exist, are all nonempty, uncountable and negligible for $\mu_p$.
