158 CHAPTER 6. MEASURES AND MEASURABLE FUNCTIONS

30. ↑Having defined the integral of nonnegative simple functions in the above problem,letting f be nonnegative and measurable. Define∫

f dµ ≡ sup{∫

sdµ : 0≤ s≤ f ,s simple}.

Show that if fn is nonnegative and measurable and n→ fn (ω) is increasing, showthat for f (ω) = limn→∞ fn (ω) , it follows that

∫f dµ = limn→∞

∫fndµ . Hint: Show∫

fndµ is increasing to something α ≤ ∞. Explain why∫

f dµ ≥ α. Now pick anonnegative simple function s ≤ f . For r ∈ (0,1) , [ fn > rs] ≡ En is increasing in nand ∪nEn = Ω. Tell why

∫fndµ ≥

∫XEn fndµ ≥ r

∫sdµ . Let n→ ∞ and show that

α ≥ r∫

sdµ . Now explain why α ≥ r∫

f dµ . Since r is arbitrary, α ≥∫

f dµ ≥ α .

31. ↑Show that if f ,g are nonnegative and measurable and a,b≥ 0, then∫(a f +bg)dµ = a

∫f dµ +b

∫gdµ