One of the quickest ways to demonstrate that there exist Lebesgue measurable subsets of the real line that are not Borel measurable is to compute the cardinality of the Lebesgue $\sigma$-algebra and the Borel $\sigma$-algebra.  The former has cardinality $2^{2^{\aleph_0}}$ (it contains the power set of the Cantor set), whereas the latter has cardinality $2^{\aleph_0}$ (by the transfinite induction construction of the Borel $\sigma$-algebra).