Omega_{1} unions of null sets: Martin's Axiom

Question

Hello,

let $U$ be the assertion "The union of $\aleph_{1}$ null sets of reals is null", i.e.

$U$ = Given any $\omega_{1}$-sequence of null sets $X_{\alpha}$, for $\alpha<\omega_{1}$, then $\bigcup_{\alpha <\omega_{1}} X_{\alpha}$ is null.

$U$ is known to be independent of ZFC. It clearly does not hold if $CH$ holds. However it follows, for example, from $MA_{\aleph_{1}}$ (Martin's Axiom at $\aleph_{1}$).

Three useful consequences of $U$ (and therefore of $MA_{\aleph_{1}}$ as well) are listed below. Let $\Omega$ be the set of Lebesgue measurable sets of reals in [0,1] and $\mu$ the uniform measure on [0,1]. Then under $U$:

1) $\omega_{1}$-completeness of $\Omega$: for any $\omega_{1}$-sequence of disjoint measurable sets $X_{\alpha}\in \Omega$, $\bigcup_{\alpha < \omega_{1}} X_{\alpha} \in \Omega$.

2) $\omega_{1}$-continuity of $\mu$: $\bigsqcup_{\alpha <\omega_{1}} \mu (X_{\alpha})= \mu ( \bigcup_{\alpha<\omega_{1}} X_{\alpha} ) $, where $\bigsqcup$ is the join operation.

3) Every $\Delta^{1}_{2}$ set is in $\Sigma$.

Now i recently wrote a paper (in computer science) that uses $U$ as hypothesis for the main theorem. I specified that the proof is valid in $ZFC+MA_{\aleph_{1}}$. However I would like to know if:

A) Has the assertion $U$ been studied independently from $MA$? Does it have a distinguished name?

B) is U strictly weaker than $MA_{\aleph_{1}}$? I guess it is, but don't know.

C) Do you have any reference to papers/book chapters/etc where the problem $U$ is considered and analyzed?

Than you in advance,

Matteo Mio

Answer 1

These questions and many other similar questions are intensely studied in the field known as cardinal characteristics of the continuum. Some ideas are mentioned in this MO answer (and also in this MO answer and this one) and you may consult Andreas Blass's excellent survey article.

In particular, the answer to all your questions is Yes. Your statement $U$ is exactly expressing that the additivity of the null ideal is at least $\omega_2$. The additivity of an ideal is the smallest cardinal $\kappa$ for which the union of any fewer than $\kappa$ many sets in the ideal is still in the ideal. Since the union of countably many measure $0$ sets has measure $0$, the additivity of the Lebesgue null ideal is always at least $\omega_1$, but as you have observed, it is consistent with ZFC that it is larger.

Other similar cardinal characteristics would include the the additivity of the meager ideal, the covering number of the null ideal (the smallest number of null sets covering $\mathbb{R}$), the covering number of the meager ideal, and many others. These cardinal characteristics and others appear in Cichon's diagram, which exhibits and explores their relationships.

Under the Continuum Hypothesis, all these cardinal characteristics agree, and have value continuum, but when CH fails, it is possible to separate them consistently with ZFC. Martin's Axiom pushes all these cardinal characteristics to the top, with value continuum. (In particular, under MA with large continuum one can strengthen your statement $U$ to allow that the union of $\aleph_{17}$ many null sets is null, or $\aleph_{\omega^2+5}$ many, etc.) To separate the cardinals from each other, therefore, rather than merely just from $\omega_1$, one uses forcing, and the most powerful results are obtained by forcing constructions carefully designed specifically for the purpose. Indeed, the study of fundamental questions in the field of cardinal characteristics of the continuum, concerning the problems of separating the cardinal characteristics, led to important advances in the theory of forcing, such as a much deeper understanding of forcing iterations and perhaps even to the rise of proper forcing.