The expansion of self-similar functions in the Faber–Schauder system

Let $\Omega = {\mathcal A}^{{\mathbb N}}$ be a space of right-sided infinite sequences drawn from a finite alphabet ${\mathcal A} = \{0,1\}$, ${\mathbb N} = \{1,2,\dots \} $. Let
$$ \rho(\mathbf{x},\mathbf{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} $$
be a metric on $\Omega = {\mathcal A}^{{\mathbb N}}$, and $\mu$ the Bernoulli measure on $\Omega$ with probabilities $p_0,p_1>0$, $p_0+p_1=1$. Denote by $B(\mathbf{x},\omega)$ an open ball of radius $r$ centered at $\mathbf{\omega}$. The main result of this paper is
$$ \mu\left(B(\mathbf{\omega},r)\right) = r+\sum_{n=0}^{\infty}\sum_{j=0}^{2^n-1}\mu_{n,j}(\mathbf{\omega})\tau(2^nr-j), $$
where $\tau(x) =2\min\{x,1-x\}$, $0\leq x \leq 1$, ($\tau(x) = 0$, if $x<0$ or $x>1$),
$$ \mu_{n,j}(\mathbf{\omega}) = \left(1-p_{\omega_{n+1}}\right) \prod_{k=1}^n p_{\omega_k\oplus j_k},\ \ j = j_12^{n-1}+j_22^{n-2}+\dots+j_n. $$
The family of functions $1,x,\tau(2^nr-j)$, $j =0,1,\dots,2^n-1$, $n=0,1,\dots$, is the Faber–Schauder system for the space $C([0, 1])$ of continuous functions on $[0, 1]$. We also obtain the Faber–Schauder expansion for the Lebesgue's singular function, Cezaro curves, and Koch–Peano curves.