  While finite non-commutative operator systems lie at the foundation of
quantum measurement, they are also tools for understanding geometric iterations
as used in the theory of iterated function systems (IFSs) and in wavelet
analysis. Key is a certain splitting of the total Hilbert space and its
recursive iterations to further iterated subdivisions. This paper explores some
implications for associated probability measures (in the classical sense of
measure theory), specifically their fractal components.
  We identify a fractal scale $s$ in a family of Borel probability measures
$\mu$ on the unit interval which arises independently in quantum information
theory and in wavelet analysis. The scales $s$ we find satisfy $s\in
\mathbb{R}_{+}$ and $s\not =1$, some $s <1$ and some $s>1$. We identify these
scales $s$ by considering the asymptotic properties of $\mu(J) /| J| ^{s}$
where $J$ are dyadic subintervals, and $| J| \to0$.
