How can i prove that the cardinality of Borel sets is less than the cardinality of lebesgue measurable sets?
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Marios Gretsas
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In this link you could find a proof that the cardinality of the collection of Borel sets $\mathcal{B}$ is the same as the cardinality of the reals.
Cardinality of Borel sigma algebra
On the other hand, every subset of a measure-zero set if Lebesgue measurable. Therefore, using the Cantor set $\mathcal{C}$ (which has the same cardinality as $\mathbb{R}$ and measure zero), we have that the cardinality of the collection $\mathcal{M}$ of Lebesgue measurable sets is at least $$|\mathcal{M}|=|\mathcal{P}(\mathcal{C})|>|\mathcal{C}|=|\mathbb{R}|=|\mathcal{B}|.$$
