Is there a probability measure on the Cantor set?

Question

I know that the Lebesgue measure of the Cantor set is $0$.

Is there a finite positive regular measure on the Cantor set?

Answer 1

Consider the intervals $I_{i_1\cdots i_n}$ of length $1/3^n$ at the $n$th level of the construction of Cantor set, and define a measure $\mu$ by $\mu(I_{i_1\cdots i_n})=1/2^n$.

Answer 2

The example in the answer of @Jonas generalizes to a central concept in probability and dynamics: the Bernoulli measures. Their description depends on the fact that the Cantor set is homeomorphic to $C = \{0,1\}^{\mathbb{N}}$ (using the product topology, starting with the discrete topology on $\{0,1\}$).

(In case this homeomorphism is unfamiliar, for each point $p$ in the middle thirds Cantor set, its "coordinates" $x_1,x_2,x_3,\ldots$ which define its image in $C$ are: $x_n=0$ or $1$ depending on whether $p$ is in the left or right half of the level $n$ subinterval containing $p$.)

Consider the "cylinder sets" $$C_n(i) = \{x \in C \,\bigm|\, x_n=i\}, \quad n \in \mathbb{N}, \quad i \in \{0,1\} $$ Choose $p \in (0,1)$ and let $q=1-p$ (in the answer of @Jonas one takes $p=\frac{1}{2}$). For each $n \in \mathbb{N}$, put measure $p$ on the cylinder set $C_n(0)$, put measure $q$ on the cylinder set $C_n(1)$. Assume that those measures are independent for different values of $n$; that allows you to extend the measure naturally to finite intersections of cylinder sets. Finally, there is a still further natural extension to all Borel subsets of $C$. The resulting class of measures on $C$, one for each $p \in (0,1)$, are called the Bernoulli measures.

One can then easily imagine a still more general class of measures, letting $p$ instead depend on $n$, the cylinder set $C_n(0)$ having measure $p_n \in (0,1)$ and its complement $C_n(1)$ having measure $q_n=1-p_n$.