I'm trying to solve this following problem:
1) M $\subset$ $\mathbb{R}^n$ is a Lebesgue mesurable set. Show that for every $\varepsilon$ > 0 there is a open set U $\supset$ M such that $\lambda$(U\M) < $\varepsilon$. Remark: [ U := $\bigcup_{i=1}^\infty$ $U_i$ , $U_i$ open ] I was able to prove this.
2) Now show that: $\lambda$(M) = inf{ $\lambda$(U) : U open , U $\supset$ M } = sup { $\lambda$(A) : A closed, A $\subset$ M } = sup {$\lambda$(K) : K compact, K $\subset$ M }
I've already proved the first equality. But I don't have any idea how to prove the other equalities. for inf{ $\lambda$(U) : U open , U $\supset$ M } = sup { $\lambda$(A) : A closed, A $\subset$ M } we could take a open set O and consider the complement of O ?
for sup { $\lambda$(A) : A closed, A $\subset$ M } = sup {$\lambda$(K) : K compact, K $\subset$ M } we could take A closed set and consider $A_k$ = A$\cap$ [-k,k] , k $\in$ $\mathbb{N}$ ?
Please help me guys. Thank you so much for your time.