Can someone help me get started on these problems? These are tough questions, and I believe some of them are propositions in some books. A reference to more info would also be nice.
[L denotes Lebesgue measure function, out denotes outer measure,
R denotes real number set]
(a)
Show that for any subset E of R, there is a G_delta set A such that E is subset of A and L(A) = out(E).
(b)
Show that a subset E of R is Lebesgue measurable if and only if there is a G_delta set A such that E is subset ofA and out(A \ E) = 0.
(c) Show that if E is subset of B, where B is a Lebesgue measurable set with L(B) < +infinity,
then
E is Lebesgue measurable if and only if L(B) = out(E) + out(B \ E)
[ How would I set up the solutions? Which should be done by contradictions? ]