I'm having trouble, please can somebody help me?
Prove that for all ε>0 there exists a closed set Fc[0,1] such that F = and λ(F)>1-ε, where λ is the Lebesgue measure.
Thank you very much!
This is an immediate consequence of the following lemma:
A set $\displaystyle X\subset \mathbb{R} ^n$ has measure $\displaystyle 0$ if and only if for all $\displaystyle \varepsilon >0$ there exists an open set $\displaystyle X\subset \Omega$ with $\displaystyle \lambda (\Omega ) <\varepsilon$
The proof of this is easy: There exists a lower semicontinous function $\displaystyle f\geq 1_{X}$ with $\displaystyle \int_{\mathbb{R} ^n} f <\frac{ \varepsilon}{2}$ then take $\displaystyle \Omega := f^{-1} \left( \frac{1}{2} , \infty \right)$