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