Differences between the Borel measure and Lebesgue measure

Not every subset of a set of Borel measure $0$ is Borel measurable. Lebesgue measure is obtained by first enlarging the $\sigma$-algebra of Borel sets to include all subsets of set of Borel measure $0$ (that of courses forces adding more sets, but the smallest $\sigma$-algebra containing the Borel $\sigma$-algebra and all mentioned subsets is quite easily described directly (exercise if you like)).

Now, on that bigger $\sigma$-algebra one can (exercise again) quite easily show that $\mu$ (Borel measure) extends uniquely. This extension is Lebesgue measure.

All of this is a special case of what is called completing a measure, so that Lebesgue measure is the completion of Borel measure. The details are just as simple as for the special case.

borel measure is defined on the smallest sigma algebra that contains all the open sets while lebesgue measure is much much more general but coincides with borel,when a set is a borel measurable.there are examples (not so easy) of non-borel lebesgue measurable sets