Consider two sets $A$ and $B$ with positive measure such that the union is $[0,1]$. Is it true that there is a one-to-one correspondence between $A$ and $B$?
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It is true without needing to assume the continuum hypothesis. A Lebesgue measurable set with positive measure has the same cardinality as $[0,1]$. My favorite proof (sketch) of this is in Ehsan's answer to a MathOverflow question. Andres Caicedo's and Fedor Petrov's answers there also address this, and another proof can be found in the expository article "Measure and cardinality" by Briggs and Schaffter. Joel David Hamkins's answer at the MathOverflow question gives a great survey of what can happen with outer measure.
I believe so, at least if we take the continuum hypothesis. This follows from the countable subadditivity of measure, for if we had a countable set we would certainly have a measure of $0$ and the union of any two countable sets is countable. So $A$ and $B$ must be uncountable and with the continuum hypothesis $|A| = |B|$. There is probably a way to do this without assuming the continuum hypothesis, but I haven't thought of it off the top of my head. Of course, even without the CH we would not be able to construct a counterexample.