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How do I prove that the class of Lebesgue measure zero sets in $\mathbb{R}^d$ has cardinality $2^{\mathfrak{c}}$ where $\mathfrak{c}$ is the power of the continuum? (I wan't to do this also for meager sets but hopefully I will deduce it from the first answer. It should be more less the same). I have the impression it is a trivial observation which I am currently missing. Any hints?

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2 Answers 2

up vote 4 down vote accepted

You just need the observation that there is one set of size continuum and Lebesgue measure zero, as the Lebesgue null sets are closed under subsets (in fact, they form an ideal). The typical example is Cantor's middle third set $C$. Any subset of it is also of measure zero, and there are $2^{|C|}=2^{\mathfrak c}$ such subsets.

For meager, the argument is very similar; again, you just need that there is a meager set of size continuum. Actually, the Cantor set works here as well.

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All subsets of the Cantor set have measure zero (and are meager); the Cantor set has cardinality $\mathfrak c$, so it has $2^{\mathfrak c}$ subsets.

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It was easy indeed. Thanks! –  Mauricio G Tec Sep 6 '13 at 21:46

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