Let' A be a subset of R such that between two its members there is a point that does not belong to A.Is A surly lebesgue measurable?If it is ,can we say it's lebesgue measure is zero.Or it is nothing to say about A.Please guide me.
Let' A be a subset of R such that between two its members there is a point that does not belong to A.Is A surly lebesgue measurable?If it is ,can we say it's lebesgue measure is zero.Or it is nothing to say about A.Please guide me.
I'm not an expert, but see this link. In particular look at property 3. My question to you would be that if you don't know if A can be measured, can you measure its compliment? If you can do so, then you can measure A as well.
-Dan
I think the Cantor set answers your question.
The set of irrational numbers satisfies the hypothesis, but its measure is not zero.
In fact, from the given hypothesis you cannot even conclude that A is measurable. Let B be any non-measurable subset of R and let A be the set of irrational numbers in B. Then A is also non-measurable, and between any two elements of A there is a rational number that is not in A.