Absolute continuity of a measure in the intervals implies absolute continuity of borel measures?
The situation is even worse than my comment. Take any measure $\nu$ that's never $0$ on a nondegenerate interval (the Lebesgue measure works, as do many finite measures). If $\nu$ has sets of measure $0$, then take one such set $E$, let $\lambda$ be a probability measure on $E$, and let $\mu = \nu + \lambda$. Then $\mu$ and $\nu$ satisfy the condition on intervals, but $\mu$ is not absolutely continuous with respect to $\nu$.