Let E⊆R and have outer measure.Show that there is an Fσ set F and a Gδ set G Such that F⊆E⊆G and m*(F)＝m*(E)=m*（G）. I can prove there is G Such that E⊆G and m*(E)=m*(G) by definition of outer measure to get ∪Iₙ and E⊆∪Iₙ, Such that m*(∪Iₙ) <m*(E)+1/n , ∪Iₙ is open take intersection of ∪Iₙ we get G but I can't find closed set to do it.
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Maybe the fact that the complement of a measurable set is also measurable might help? – GSofer Sep 20 '20 at 12:39
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Please use MathJax to render the math Thank you. – user577215664 Sep 20 '20 at 12:39
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But E isn't a measurable set – user793548 Sep 21 '20 at 04:22
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Sorry ,but i don't kown use MathJax by phone .I haven't computer – user793548 Sep 21 '20 at 04:28