418 CHAPTER 15. THE Lp SPACES

15. Suppose f ,g≥ 0. When does equality hold in Holder’s inequality?

16. Prove Vitali’s Convergence theorem: Let { fn} be uniformly integrable and complexvalued, µ(Ω) < ∞, fn(x)→ f (x) a.e. where f is measurable. Then f ∈ L1 andlimn→∞

∫Ω| fn− f |dµ = 0. Hint: Use Egoroff’s theorem to show { fn} is a Cauchy

sequence in L1 (Ω). This yields a different and easier proof than what was doneearlier. See Theorem 11.5.3 on Page 257.

17. ↑ Show the Vitali Convergence theorem implies the Dominated Convergence theo-rem for finite measure spaces but there exist examples where the Vitali convergencetheorem works and the dominated convergence theorem does not.

18. ↑ Suppose µ(Ω)< ∞, { fn} ⊆ L1(Ω), and∫Ω

h(| fn|)dµ <C

for all n where h is a continuous, nonnegative function satisfying

limt→∞

h(t)t

= ∞.

Show { fn} is uniformly integrable. In applications, this often occurs in the form ofa bound on || fn||p.

19. ↑ Sometimes, especially in books on probability, a different definition of uniformintegrability is used than that presented here. A set of functions, S, defined on afinite measure space, (Ω,S ,µ) is said to be uniformly integrable if for all ε > 0there exists α > 0 such that for all f ∈S,∫

[| f |≥α]| f |dµ ≤ ε.

Show that this definition is equivalent to the definition of uniform integrability givenearlier in Definition 11.5.1 on Page 256 with the addition of the condition that thereis a constant, C < ∞ such that ∫

| f |dµ ≤C

for all f ∈S.

20. f ∈ L∞(Ω,µ) if there exists a set of measure zero, E, and a constant C < ∞ such that| f (x)| ≤C for all x /∈ E.

|| f ||∞ ≡ inf{C : | f (x)| ≤C a.e.}.

Show || ||∞ is a norm on L∞(Ω,µ) provided f and g are identified if f (x) = g(x) a.e.Show L∞(Ω,µ) is complete. Hint: You might want to show that [| f |> || f ||

∞] has

measure zero so || f ||∞

is the smallest number at least as large as | f (x)| for a.e. x.Thus || f ||

∞is one of the constants, C in the above.