The Cantor set has dimension log 2 / log 3 and critical measure one

Given: The objects, conventions, and hypotheses in the statement above.

Under the standing Countable Choice hypothesis, for the Cantor probability measure, every nonempty bounded set $U$ has ${\mu }_c^{\ast }(U)\le (\mathrm{diam}U{)}^s$ at $s=\log 2\mathrm{/}\log 3$. The sharp interval bound for Cantor measure

Under the standing Countable Choice hypothesis, a finite Borel measure with outer mass on $A$ positive and diameter bound $C_0{r}^s$ gives ${\mathcal{H}}^s(A)\ge {\mu }^{\ast }(A)\mathrm{/}{C}_0$. The mass distribution principle

Finite positive Hausdorff measure at $s$ forces dimension $s$. Hausdorff dimension is the unique critical exponent

$C$ consists exactly of the ternary expansions using only zero and two. The Cantor set is exactly the set of ${\sum }_{k\ge 1}{a}_k{3}^{-k}$ with every $a_k\in \{0,2\}$, and this gives a bijection with $\{0,1{\}}^{\mathbb{N}}$

Positive real powers obey the multiplication and power-of-power laws. The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

Under the standing Countable Choice hypothesis, ${\mu }_c$ is a probability measure concentrated on $C$. The Cantor measure is a singular atomless probability measure concentrated on the Cantor set

Under Countable Choice, ${\mu }_c={\mu }_{F_c}$, where the extended Cantor function $F_c$ is nondecreasing and right-continuous on $\mathbb{R}$. The Cantor measure

Under Countable Choice, the Lebesgue–Stieltjes measure ${\mu }_F$ of a nondecreasing right-continuous real function $F$ is a Borel measure on $\mathbb{R}$. Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $\mathbb{R}$