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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).