A set E in [a, b] is said to be (Lebesgue) measurable if its characteristic function 1E (defined by 1ε(x) := 1 if x ∈ E and 1ε(x) := 0 if x ∈ [a, b]\E) belongs to M[a, b]. We will denote the collection of measurable sets in [a, b] by M [a, b]. In this exercise, we develop a number of properties of M [a, b].
(a) Show that E ∈ M [a, b] if and only if 1E belongs to R*[a, b].
(b) Show that θ ∈ M [a, b] and that if [c, d] ⊂ [a, b], then the intervals [c, d], [c, d), (c, d], and (c, d) are in M [a, b].
(c) Show that E ∈ M [a, b] if and only if Eʹ:= [a, b]\E is in M [a, b].
(d) If E and F are in M [a, b], then E ⋃ F, E ∩ F and E\F are also in M [a, b]. [Show that 1E⋃F = max{1E, 1F}, etc.]
(e) If (Ek) is an increasing sequence in M [a, b], show that E := ⋃∞k=1 Ek is in M [a, b]. Also, if (Fk) is a decreasing sequence in M [a, b], show that F := ∩∞k=1 Fk is in M [a, b].
(f) If (Ek) is any sequence in M [a, b], show that ⋃∞k=1 Ek and ∩∞k=1 Ek are in M [a, b].