[!]. This exercise is used in Section 12.3. Let f,g be bounded on [a,bJ.

(

a) Prove that

(U) lb (f(x) + g(x)) dx :::; (U) lb f(x) dx + (U) lb g(x) dx and

(L) lb (f(x) + g(x)) dx ~ (L) lb f(x) dx + (L) lb g(x) dx.

(b) Prove that

(U) lb f(x) dx = (U) lc f(x) dx + (U) lb f(x) dx and

(L) lb f(x) dx = (L) l c f(x) dx + (L) lb f(x) dx for a < c < b.

~. This exercise is used in Sections e5.5, 6.2, and e7.5.