This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.

**Question 1.** Does every set of reals contain a measure-zero subset
of the same cardinality?

In other words, if $A\subset\mathbb{R}$, is there a measure-zero set $B\subset A$ with $|B|=|A|$? Is this assertion at least consistent? Does it follow from the continuum hypothesis? Does it follow from some other cardinal characteristic hypothesis? In the intended application, what is needed is that the assertion is consistent with the additivity number for measure being equal to the continuum. Is this consistent? Can anyone prove the consistency of the failure of the property?

Similarly, in the case of category rather than measure:

**Question 2.** Does every set of reals contain a meager subset of the same cardinality?

And similarly, is this statement consistent? Does it follow from CH or other cardinal characteristic hypotheses? Is it consistent with the additivity number for the meager ideal being large? Can anyone show the consistency of the failure of the property?

The questions arise in my post on Ashutosh's question, where I had proposed as a solution idea the strategy of a back-and-forth construction of length continuum, where the domain and target remain measure-zero during the course of the construction. But in order for this strategy to succeed, we seem to need to know in the context there that one may extend a given measure-zero set inside another non-measure-zero set to a larger measure-zero set with the same cardinality (and the same with meagerness). I had thought at first that this should be easy, but upon reflection I am less sure about it, and so I ask these questions here.