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.
Best Answer
It is consistent that both of the questions have a negative answer. Indeed, this happens if MA holds.
A set $E$ of reals is called a Luzin set if $E$ has size continuum and for every meager set $X$ the intersection $E\cap X$ has size less than continuum.
A set of reals $E$ is called a Sierpiński set if $E$ has size continuum and for every measure zero set $X$ the intersection $E\cap X$ has size less than continuum.
Theorem: MA implies that there are Luzin and Sierpiński sets.
Proof: To construct a Luzin set, list all Borel nowhere dense sets in order type continuum: $\langle F_\alpha;\alpha<\mathfrak{c}\rangle$. For each $\alpha$ choose some $e_\alpha\notin \bigcup_{\beta<\alpha}F_\beta$; this is possible since MA implies that the union of less than continuum meager sets is meager. $E=\{e_\alpha;\alpha<\mathfrak{c}\}$ has size continuum and its intersection with every closed nowhere dense set has size less than continuum by construction. But since a meager set is contained in a union of countably many closed nowhere dense sets, $E$ must be a Luzin set.
To construct a Sierpiński set, replace "Borel nowhere dense" above by "Borel of measure zero" and "meager" by "measure zero". $\square$
In particular, this shows that assuming cardinal characteristics are large is not helpful for this problem.
The same avoidance idea seems to also show:
Theorem: If $V$ was obtained from $W$ by adding more than $\mathfrak{c}^W$ many Cohen (or random) reals to $W$, then the set of generic reals is Luzin (or Sierpiński) in $V$.