What is the Lebesgue outer measure of a Vitali set and its complement over $\Omega=[0,1]$?
My first guess was zero and one but that was on my wrong idea that I can adjust the vitali set to lie within $[0,\varepsilon)$ changing it again and again… Also though its complement lies within $[\varepsilon,1]$ it is not guaranteed that there's a finer covering of it than that of $[\varepsilon,1]$.
Best Answer
As is noted here:
What is the outer measure of Vitali set?
You may change the vitali set to have any possible Outer Measure if you construct it properly.
In general, it is not Lebesgue Measurable.