Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment VIII.

1. Problem 1. a) Let μ be a measure on X, f : X → [0, ∞] be measurable, ∫_{X} f(x)dμ = c, where 0 < c < ∞, and α > 0 be a constant. Find

lim_{n→∞} ∫_{X} log (1 + (\frac{f(x)}{n})^{α})^{n}dμ.

Hint: If α ≥ 1, the integrands are dominated by αf(x). If 0 < α < 1, Fatou’s lemma can be applied.

b) Put f_{n}(x) := 1_{A}(x) if n is odd, and f_{n}(x) := 1−1_{A}(x) if n is even. Show that strict inequality can occur in Fatou’s lemma.

c) It is easy to guess the limits of

∫_{0}^{n} (1 − \frac{x}{n})^{n}e^{x/2}dx,  ∫_{0}^{n} (1 + \frac{x}{n})^{n}e^{−2x}dx

as n → ∞. Prove that your guesses are correct.

d) Does

lim_{n→∞} \frac{1}{n} ∫_{0}^{1} log (1 + e^{nf(x)})dx

exist for every real f ∈ L([0, 1], dx)? If it exists, what is it?

2. Problem 2. a) Let μ(X) < ∞, (f_{n})_{n=1}^{∞} be a sequence of bounded measurable functions on X. Assume also that f_{n} → f uniformly as n → ∞. Prove that

lim_{n→∞} ∫_{X} f_{n}(x)dμ = ∫_{X} f(x)dμ,

and show that the hypothesis “μ(X) < ∞” can not be omitted. Hint: Take X = R, μ - Lebesgue measure, f_{n}(x) = 1_{[−n^{2},n^{2}]}(x)/n.

Definition. Let μ be a measure on X. We say that a sequence of integrable functions (f_{n})_{n=1}^{∞} converges to integrable f in L^{1}(X, μ)-sense, if for any ϵ > 0 there exist a natural N = N(ϵ), such that ∀n > N we have

∫_{X} |f_{n}(x) − f(x)|dμ < ϵ.

b) Construct a sequence of integrable on [0, 1] functions (f_{n})_{n=1}^{∞} satisfying

lim_{n→∞} f_{n}(x) = 0 ∀x ∈ [0, 1], ∫_{[0,1]} |f_{n}(x)|dx ≤ C ∀n,

but such that the sequence does not converge in L^{1}([0, 1], dx). Hint: Consider f_{n}(x) := n1_{(0,1/n](x)}.

c) Construct a sequence (f_{k})_{k=1}^{∞} of continuous on [0, 1] functions, such that

0 ≤ f_{k}(x) ≤ 1,  lim_{k→∞} ∫_{[0,1]} f_{k}(x)dx = 0,

but such that the sequence converges for no x ∈ [0, 1]. Hint: Consider f_{k}(x) defined in Assignment VI, Problem 4, b).

d) Let (f_{n})_{n=1}^{∞} be a sequence of nonnegative integrable functions on X converging to f almost everywhere as n → ∞. Prove that

∫_{X} f_{n}(x)dμ → ∫_{X} f(x)dμ as n → ∞

implies f_{n} → f in L^{1}(X, μ) as n → ∞. Hint: Let ∫_{X} f(x)dμ := A. For any ϵ > 0 ther exists a subset E_{1} ⊂ X of finite measure such that

∫_{E_{1}} f(x)dμ > A − ϵ.

There exists δ = δ(ϵ) > 0, such that

∫_{E} f(x)dμ < ϵ ∀E : μ(E) < δ.

Moreover, there exists E_{2} ⊂ E_{1} such that μ(E_{1} \ E_{2}) < δ, and f_{n} converges uniformly on E_{2}. We have

∫_{X} |f_{n}(x) − f(x)|dμ = ∫_{E_{2}} |f_{n}(x) − f(x)|dμ + ∫_{X\E_{2}} |f_{n}(x) − f(x)|dμ = ...

3. Problem 3. Let E_{i}, i = 1, ..., n be measurable subsets of [0, 1] such that every point x ∈ [0, 1] belongs to at least q sets E_{i}, q ≤ n. Prove that there exists i such that m(E_{i}) ≥ q/n.

4. Problem 4. Let f(x) be bounded on [0, 1]. Prove that ∫_{0}^{c} f(x)dx = 0 ∀c ∈ [0, 1] implies

f(x) = 0 almost everywhere.

Hint:

a) Observe that for any open interval (a, b) ⊂ [0, 1] we have

∫_{0}^{b} f(x)dx − ∫_{0}^{a} f(x)dx = 0,

and the same is true for any open set.

b) Prove that the same is true for any closed subset of [0, 1].

c) Prove that the same is true for any subset X := ∪_{i=1}^{∞}F_{i} ⊂ [0, 1] of type F_{σ} (a set X is called of type F_{σ} if it is a union of countably many closed sets). To do this, assume at first that F_{i} ⊂ F_{i+1} (otherwise put F_{2} = F_{1} ∪F_{2}, F_{3} = F_{1} ∪F_{2} ∪F_{3},..., F_{i} = ∪_{k+1}^{i}F_{k}). Then observe that

X = F_{1} ∪ (F_{2} \ F_{1}) ∪ (F_{3} \ F_{2}) ∪ ... ∪ (F_{i+1} \ F_{i}) ∪ ...