Characterization of a probability measure

Question

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?

Answer 1

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 $\mathcal{B}$.