Can a meager set have positive measure?
I know every meager set (contained in a a countable union of nowhere dense sets) of the real line has an empty interior. But is it possible that a meager set has positive Lebesgue measure?
Sets of positive measre can be meager, right? Do you have an example?
Thank you
Yes,a meager set can have positive - even infinite -measure. Maybe this is not a good example, but that's what came to my mind:
Let r_n be an enumeration of the rationals and let a_n be a sequence of positive numbers such that Sum a_n converges. Let A = Union [(r_n - a_n , r_n + a_n)]. Then, A is open (union of open intervals), is dense in R (contains the rationals) and is not all of R, because it has finite measure and m(R) = oo. Actually, the subadditivty of the measure implies that
m(A) <= Sum (n=1, oo) m((r_n - a_n, r_n + a_n)) = Sum (n=1, oo) 2 a_n < oo, since Sum (n=1, oo) converges.
Now, let B be the complement of A. Then, B is closed, has an empty interior (because A is dense) and has infinite measure. Hence, B is not only meager, but nowhere dense, and has infinite measure.
If you want finite positive measure, do something similar enumerating the rationals in [0, 1].
This answers your second question, but I guess what you really want to ask is if sets of measure zero (null sets) need to be meager. Right? I don't have an example now, but, if my guess is right, the answer is no. I'm not perfectly aware of their construction, hope someone else can help you, but I know there are some sets known as "fat Cantor sets". Their measure is 0 but they are not meager.
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