Integration Theory / Mathematical Analysis (5B1452/MA429)

Homework assignment # 2, due October 31

1. Which of the functions below are Lebesgue-integrable? a. f(x) = sin(x) + cos(x) on the whole real line. b. f(x) = 1/x^{2} on the interval [0, 1]. c. f(x) = 1/x^{2} on the interval [1, +∞].

2. Find the following limits:

a.  lim_{n→∞} ∫_{0}^{1} \frac{1 + nx}{(1 + x)^{n}} dx .

b.  lim_{n→∞} ∫_{0}^{n}(1 + x/n)^{n}e^{−2x}dx .

Hint:Use Lebesgue bounded or monotone convergence theorems.

3. Let {g_{n}} be a monotone decreasing sequence of non-negative measurable functions, converging to a function g. a. Prove, that if g_{1} is integrable, then all g_{n} and g are integrable and

lim_{n→∞} ∫ g_{n}dμ = ∫ gdμ . (⋆)

b. Construct a counterexample to (⋆), with g_{1} non-integrable (i.e. ∫ g_{1} = +∞).

4. Let f(x) be a Lebesgue integrable function on the real line. a. Prove that for Lebesgue almost every real x the (biinfinite) series ∑_{n=−∞}^{+∞} f(x+n) converges absolutely. b. Prove that for a function g defined by g(x) := ∑_{n=−∞}^{+∞} f(x + n), one has

∫_{0}^{1} g(x)dx = ∫_{−∞}^{+∞} f(x)dx .

Hint: Show that it is sufficient to deal with non-negative function f, and then use theorem about summing series under the integral sign (and translation invariance of Lebesgue measure).

4. Define function f on the interval [0, 1] by the following rule: f(x) = 0 for rational x and f(x) = [1/x] for irrational x ([…] denotes the integral part). a. Show, that f is measurable. b. Show, that ∫_{0}^{1} f(x)dx = +∞.