If E M [a, b], we define the (Lebesgue) measure of E to be the number m(E) := «ba 1E. In this exercise, we develop

If E ˆˆ M [a, b], we define the (Lebesgue) measure of E to be the number m(E) := ˆ«ba 1E. In this exercise, we develop a number of properties of the measure function m : M [a, b] †’ R.
(a) Show that m(θ) = 0 and 0 (b) Show that m([c, d]) = m([c, d)) = m((c, d]) = m((c, d)) = d - c.
(c) Show that m(Eʹ) = (b - a) - m(E).
(d) Show that m(E ‹ƒ F) + m(E ˆ© F) = m(E) + m(F).
(e) If E ˆ© F = θ show that m(E ‹ƒ F) = m(E) + m(F). (This is the additivity property of the measure function.)
(f) If (Ek) is an increasing sequence in M [a, b], show that m [‹ƒˆžk=1 Ek) = limk(Ek). [Use the Monotone Convergence Theorem.]
(g) If (Ck) is a sequence in M [a, b] that is pairwise disjoint (in the sense that Cj ˆ© Ck = θ, whenever j ‰  k), show that

(This is the countable additivity property of the measure function.)

(18)