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How can one show that the Lebesgue measure is not complete on $(\mathcal{R}, \mathbf{B}(\mathcal{R}))$?

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3 Answers 3

$\mathcal B(\mathbb R)$ has the cardinality of the continuum. Cantor set too and moreover it has measure $0$. Hence there's at least a subset of Cantor set which is not measurable.

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How: You would have to give an example of a subset of measure zero with a subset which is not measurable.

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Construct a subset $C$ of measure zero, and $|C|=|R|$, notice that the set of Borel sets has the same cardinality as $R$.

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