# Characterization of a probability measure

Let $$\mathbb{P} : \mathcal{B} \to [0,1]$$ be a real probability measure, where $$\mathcal{B}$$ is the Borel $$\sigma-$$algebra. Given the specific form of $$\mathbb{P}$$, we are asked to show that $$\mathbb{P}$$ is (absolutely) continuous. The way I'd proceed is to calculate the probability of a sub-interval $$[a,b]$$, writing it in terms of the integral of a density $$f:\mathbb{R}\to [0,\infty)$$. However, sub-intervals are not the whole Borel $$\sigma-$$algebra, so shouldn't I do the same calculations as above for any Borel set? Is there some theorem that states that calculating the probability on sub-intervals is enough to characterize the probability uniquely?

• If $\mathbb P (A)=\int_A f(x)dx$ for every interval $A=[a,b]$ (where $f$ is a non-negative measurable function) then the same holds for every Borel stet $A$. Apr 7, 2023 at 11:18

## 1 Answer

The theorem you are looking for is the Carathéodory Extension theorem. If two probability measures agree on the set of intervals in $$\mathbb{R}$$, then they agree on the $$\sigma$$-algebra generated by the intervals, which is $$\mathcal{B}$$.

• Carathéodory does ring a bell. Thanks a lot. Apr 7, 2023 at 11:29