What mistake is in my "proof" that the set of all subsets of R with lebesgue measure 1 has a maximal element?

Yes, that's the problem. The chains are not countable, so there is no reason for $A $ to be measurable. The Corollary also requires a countable (and not arbitrary) intersection.


A "chain" is just any totally ordered subset of $K$. It doesn't have to be a sequence indexed by the natural numbers; it might be uncountable. So you can't assert that $m(A) = \lim_{i\rightarrow \infty} m(A_i) = 1$, since you don't necessarily have a sequence at all.

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Measure Theory

Lebesgue Measure

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