Integration Theory / Mathematical Analysis (5B1452/MA429)

Homework assignment # 4

This assignment is due Thursday, December 5th. No late homework will be accepted.

Final exam will be on December 12th, 10:00-14:00, in classrooms K51 and K53. It will be written, and will contain a few problems similar to those in the homework assignments.

1.a. Show that f(x, y) = 1/\sqrt{x^{2} + y^{2}} is Lebesgue-integrable in the square [0, 1] × [0, 1]. b. Evaluate ∫_{0}^{1} ∫_{0}^{1} \frac{x−y}{(x+y)^{3}} dxdy and ∫_{0}^{1} ∫_{0}^{1} \frac{x−y}{(x+y)^{3}} dydx. Why Fubini theorem does not apply?

2. Let B_{j} and L_{j} denote Borel (i.e. generated by open sets) and Lebesgue (i.e. of Lebesguemeasurable sets) σ-algebras in ℝ^{j} correspondingly. a. Show that B_{1} × B_{1} = B_{2} b. Show that L_{1} × L_{1} ≠ L_{2}. Hint: Show that every subset of any line in the plane belongs to L_{2}, but not every – to L_{1} ×L_{1}. 3.a. Consider some measure space (X, A, μ) with finite measure μ. Show that if 1 ≤ p ≤ r ≤ ∞, then L^{p}(X, μ) ⊃ L^{r}(X, μ). b. Show that the latter is no longer true for infinite measures μ.

4. Show that translation is continuous in L^{1}(ℝ), i.e. that for f ∈ L^{1}(ℝ) one has

lim_{ϵ→0} ∥f(x) − f(x + ϵ)∥_{1} = 0

Hint: Prove first for continuous functions, which are zero outside interval [a, b]. Then prove for L^{1} functions, which are zero outside [a, b] (approximating them by continuous – Lusin’s theorem), and finally for general L^{1} functions.

5. Convolutions. Convolution h = f ∗g of two Lebesgue measurable on the real line functions f, g is defined by

h(x) := ∫_{ℝ} f(x − y)g(y)dy ,

provided this integral exist. a. Show that if f ∈ L^{1}(ℝ) and g ∈ L^{1}(ℝ), then f ∗ g ∈ L^{1}(ℝ) (particularly, f ∗ g is well-defined and a.e. finite). b. Show that f ∗ g = g ∗ f c. Show that if f ∈ L^{1}(ℝ) and g is continuous, zero outside some finite interval [a, b], then f ∗ g is continuous.