Independence of the statement that a union of $<\mathfrak{c}$ many measure zero sets has measure zero I read that that the statement that a union of fewer than continuum many measure zero sets has measure zero is independent of $\sf{ZFC}$. Is there a standard reference to this result?
 A: I think the book by Bartoszyński and Judah is one of the standard references for this kind of result (especially, cardinal characteristics of the continuum that are related to null and meager ideals.) Let me sketch how you can construct a proof.
On the one hand, Martin's axiom $\mathsf{MA}$ proves that the union of $<\mathfrak{c}$ null sets is always null. You can see the proof of the consistency of Martin's axiom in some textbooks about forcing (if it covers iterated forcing. Jech and Kunen certainly mention it.)
(Note. I realized that Martin's axiom is overkill. Just assuming the continuum hypothesis suffices. However, Martin's axiom shows that the union of $<\mathfrak{c}$ null sets is still null regardless of what the value of $\mathfrak{c}$ is.)
On the other hand, establishing the consistency of the negation of your statement is usually stated in the following combinatorial form: can we force the following cardinal
$$\mathsf{add}(\mathcal{N}):= \min\{\kappa : \text{the union of some $\kappa$ null sets is not null}\}$$
strictly less than $\mathfrak{c}$? Section 7.3.A of Bartoszyński and Judah discusses how to force $\mathsf{add}(\mathcal{N})<\mathfrak{c}$.
