While reading Robert Ash's Basic Probability Theory, I saw the following theorem (p.53):
Let $f$ be a nonnegative real-valued function on $\mathbb{R}$, with $\int_{-\infty}^\infty f(x)\,dx=1$. There is a unique probability measure $P$ on the Borel subsets of $\mathbb{R}$, such that $P(I)=\int_If(x)\,dx$ for all intervals $I$ in $\mathbb{R}$.
My question is this: If there is no such density function $f(x)$, then is it possible that a probability measure on Borel subsets of $\mathbb{R}$ is not completely determined by its values on intervals?
Finite disjoint unions of ingtervals of the type $[a,b)$ form an algebra which generates the Borel sigma algebra. This implies that if two probability measures agree on intervals they agree on all Borel sets.