# Examples of the Carathéodory extension theorem

What are examples of the Carathéodory Extension Theorem that is not Lebesgue measure or Hausdorff measure?

One very important example is the construction of product measures. Given measures $\mu$ and $\nu$ on measurable spaces $X$ and $Y$, we seek to define a product measure $\mu\times \nu$ on $X\times Y$. Now, it is easy to define what this product measure should be on rectangles: if $A\subseteq X$ and $B\subseteq Y$ are measurable, then $(\mu\times\nu)(A\times B)$ should be $\mu(A)\nu(B)$ (just like the usual formula for the area of a rectangle in the plane). This can then be extended to finite unions of rectangles, and then the Caratheodory extension theorem can be used to extend it to a measure defined on the entire $\sigma$-algebra generated by such rectangles.
Let $P$ be a probability measure on $(\mathbb{R}, \mathcal{B}\mathbb{R})$. The cdf of $P$ is the function $$F: \mathbb{R} \rightarrow [0,1], x \mapsto P((-\infty, x]).$$ Given the measure $P$, one can obviously construct $F$, but $F$ can also be used to reconstruct $P$:
Given a cdf $F$. The according probability measure $P$ is the unique measure on $(\mathbb{R}, \mathcal{B}\mathbb{R})$, such that $P((a,b]) = F(b)-F(a)$. One can prove that $P$ exists and that it is unique, using the Carathéodory Extension Theorem.