Finitely additive, $\kappa$-additive atomless measures in ZFC

Under Martin's Axiom (and non-CH) the Lebesgue measure is $$2^\omega$$-additive in the sense that unions of fewer than continuum ($$2^\omega$$) many null sets are measureable and null. In ZFC we may however extend the Lebesgue measure to a finitely-additive measure on the power set of $$[0,1]$$ and still call it atomless.

Are there ZFC examples of finitely-additive measures that extend an atomless measure $$\mu$$ and are $$\kappa$$-additive in the above sense w.r.t $$\mu$$-null sets, where $$\kappa > 2^\omega$$?

I claim that a non-atomic measure $$\mu$$ can never be $$<{2^\omega}^+$$-additive. Then the same applies to any finitely-additve extension.

Let $$(\Omega, \frak{A}, \mu)$$ be a measure space and let us assume that $$\mu$$ is non-atomic. It follows that there exists $$A \in \frak{A}$$ such that $$0 < \mu(A) < \infty$$. I now want to partition $$A$$ into $$2^\omega$$ many null sets. Start by splitting $$A$$ into $$A_0$$ and $$A_1$$ both are sets in $$\frak{A}$$ of positive measure. This can be done, since $$\mu$$ is non-atomic. Assume that $$A_s$$ for $$s\in 2^{<\omega}$$ has been defined and partition it into $$A_{s^\frown 0}$$ and $$A_{s^\frown 1}$$. For every $$x \in \,^{\omega}2$$ define $$A_x:= \bigcap_{n < \omega} A_{x \restriction n}$$, which is the first limit step.

First note that $$\mu(A_x)=\inf_{n < \omega} \mu(A_{x \restriction n})$$ and that some $$A_x$$ may already have measure 0, while others may still have positive measure. If $$A_x$$ has measure 0, then we do not have to take care of it anymore. If $$A_x$$ has positive measure, we continue as before and split it into $$A_{x^\frown 0}$$ and $$A_{x^\frown 1}$$. This way we get a (transfinite) binary tree, such that some branches die out at limit stages.

I claim that every branch dies out at countable height, i.e. there is no $$\omega_1$$-branch. If not, there exists a $$\subseteq$$-decreasing sequence $$(A_\alpha)_{\alpha < \omega_1}$$ of length $$\omega_1$$ such that $$\mu(A_\beta) < \mu(A_\alpha)$$ if $$\beta > \alpha$$. But this is impossible, since there cannot exist an uncountable decreasing sequence in $$\mathbb{R}$$ (separability). Therefore, there are only $$2^\omega$$ branches and so $$A=\bigcup_{\alpha < 2^\omega} A_\alpha$$.

This and (much) more can be found in Jech's book in chapter 10.

• Right, so it seems the right question is whether there is a ZFC example of an atomless $(2^\omega)$-additive measure?
– Kant
May 6, 2020 at 5:18
• Let me ask a follow-up question.
– Kant
May 6, 2020 at 5:31