A set E in [a, b] is said to be (Lebesgue) measurable if its characteristic function IE

Chapter 10, Problem 15

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A set E in [a, b] is said to be (Lebesgue) measurable if its characteristic function IE (definedby 1E(x) := 1 if x E E and 1E(x) := 0 if x E [a, b] \ E) belongs to M[a, b]. We will denotethe collection of measurable sets in [a, b] by M[a, b]. In this exercise, we develop a number ofproperties ofM[a, b].(a) ShowthatE E M[a, b]if andonlyif IE belongsto R*[a, b].(b) Show that 0 E M[a, b] and that if [e, d] S; [a, b], then the intervals [e, d], [e, d), (e, d]and (e, d) are in M[a, b].(c) Show that E E M[a, b] if and only if E' := [a, b] \ E is in M[a, b].(d) If E and F are in M[a, b], then E U F, E n F and E \ F are also in M[a, b]. [Hint: Showthat 1EUF= max{lE, IF}' etc.](e) If (Ek) is an increasing sequence in M[a, b], show that E := U~I Ek is in M[a, b]. Also,if (Fk) is a decreasing sequence in M[a, b] show that F := n~1 Fk is in M[a, b]. [Hint:Apply Theorem IOA.9(c).](f) If (Ek) is any sequence in M[a, b], show that U~I Ek and n~1 Ek are in M[a, b].