Examples of the Carathéodory extension theorem

Question

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

Answer 1

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.

Answer 2

The concept of a cumulative distribution functions (cdf) is closely related to Carathéodory Extension Theorem. The construction is fairly similar to the Lebesgue measure though.

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.