# Is the Lebesgue $\sigma$-algebra bigger than the Borel $\sigma$-algebra without axiom of choice?

It is wellknown, that using the axiom of choice the Borel $\sigma$-algebra has cardinality $2^\mathbb{N}$, whereas the Lebesgue $\sigma$-algebra has cardinality $2^\mathbb{R}$. It immediately follows, that there are (many) Lebesgue-measurable sets, which are not Borel-measurable.

Now I know, that the existence of avmeasure on $\mathcal{P}(\mathbb{R})$ is consistent with ZF. Though this really doesn't say anything about the Borel algebra in the absence of choice it lead me to the question if $Bor_\mathbb{R}=Leb_\mathbb{R}$, where the first is the smallest $\sigma$-algebra containing all intervals and the second are the measurable sets of the (outer) Lebesgue measure.

Are there any sources on this?

• Without the Axiom of Choice, you can prove that the Cantor set is an uncountable Borel subset of $[0,1]$ with Borel measure $0$. Any subset of the Cantor set is thus Lebesgue-measurable, and we know that there are $2^{\mathbb{R}}$-many of these. – Berrick Caleb Fillmore Oct 9 '17 at 22:06
• Yes, but can you prove without Choice, that the Borel-algebra has cardinality $2^\mathbb{N}$? The proof I know uses transfinite induction on the Borel hierarchy and I think the latter one need choice. – Takirion Oct 9 '17 at 22:24
• mathoverflow.net/questions/32720/… looks to be somewhat relevant here... – Steven Stadnicki Oct 9 '17 at 22:31
• @Takirion: There is no such measure. Any measure in such a model would have to be the trivial measure as a countable subset of $\mathbb{R}$ has null measure (as can be seen from its outer measure). – Berrick Caleb Fillmore Oct 10 '17 at 10:07
• @Takirion: The moment you assume $\mathsf{AC}_{\omega}$, it is no longer true that $\mathbb{R}$ is a countable union of countable sets (because $\mathsf{ZF} + \mathsf{AC}_{\omega}$ says that a countable union of countable sets is countable, while $\mathsf{ZF}$ says that $\mathbb{R}$ is uncountable), so measure theory becomes non-trivial. – Berrick Caleb Fillmore Oct 10 '17 at 10:12