Lebesgue measure is the completion of Borel measure, meaning if $A\subseteq B\subseteq C$ and $A$ and $C$ are Borel sets with equal measure, then $C$ is a Lebesgue-measurable set with that same measure.
If some measure on Borel sets is absolutely continuous with respect to Lebesgue measure restricted to Borel sets and Lebesgue measure restricted to Borel sets is absolutely continuous with respect to that measure, then the set of measurable sets in the completion of that measure is the same as the set of Lebesgue-measurable sets.
But probabilists consider measures on the set of all Borel sets for which that mutual absolute continuity does not happen. The simplest case is any discrete distribution, i.e. any distribution made up entirely of point masses. Another is the Cantor distribution: Suppose $X$ is a number of the form $0.d_1d_2d_3\ldots$ in base $3$, and each digit is $0$ or $2,$ with probability $1/2$ each, and all the digits are independent. This distribution does not assign positive probability to any set with only one member (so it has no discrete part) but also has no part that is absolutely continuous with respect to Borel measure.