TITLE: Is sigma-additivity of Lebesgue measure deducible from ZF?
QUESTION [11 upvotes]: Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. Follow-up question: Jech's 1973 book on the axiom of choice seems to be cited as the source for the Feferman-Levy model. Can this be sourced in the work of Feferman and levy themselves? Are these S. Feferman and A. Levy?

REPLY [11 votes]: This depends on exactly how you define Lebesgue measure since some definitions incorporate countable additivity. However, there is a model of ZF, the Feferman-Levy model, where $\mathbb{R}$ is a countable union of countable sets which ensures that any countably additive measure on $\mathbb{R}$ has to be trivial.

REPLY [9 votes]: No, you can't have that. It is consistent that the real numbers are a countable union of countable sets, in which case you immediately have that there is no nontrivial measure which is countably additive on the real numbers.
There are other models, however, in which $\aleph_1$ is singular, the countable union of countable sets of real numbers is countable; but every set is Borel. In such models, I believe, you can't have a countably additive Lebesgue measure as well.

To your question, yes. These are Solomon Feferman and Azriel Levy. The result appears as an abstract in Notices of the AMS from 1964 (give or take a year, this is from memory).