MAT 550, REAL ANALYSIS SUNY at Stony Brook, SPRING 99 Problem Set 3 Due on Friday, Febuary 12

1. For each 1 ≤ p ≤ ∞ give an example of a function f ∈ L^{p}(ℝ, dx) such that f ∉ L^{q}(ℝ, dx) for any q ≠ p.

2. Give an example of a function f ∈ L^{1}([0, 1], dx) such that f ∉ L^{2}([a, b], dx) for any interval 0 ≤ a < b ≤ 1.

3. Give an example of a function f on [0, 1] which is in every L^{p}([0, 1], dx) for every p < ∞, but which is not in L^{∞}([0, 1], dx)].

4. Show that 1 ≤ p < q < ∞ implies L^{p}(μ) ⊂ L^{q}(μ) iff there is a ϵ > 0 so that μ(E) < ϵ ⇒ μ(E) = 0 for every μ-measurable E.

5. Show that 1 ≤ p < q < ∞ implies L^{p}(μ) ⊃ L^{q}(μ) iff there for every ϵ > 0, X contains only finitely many disjoint sets of measure ϵ < μ(E) < ∞.

6. Show the following are equivalent:

(a) L^{p}(μ) = L^{q}(μ) for all 1 ≤ p < q < ∞.

(b) L^{p}(μ) is finite dimensional for all 1 ≤ p < ∞.

(c) L^{p}(μ) consists entirely of simple functions for every 1 ≤ p < ∞.

(d) X can be written as a finite union of atoms (i.e., X is the disjoint union ∪_{k=1}^{n}X_{k} where each X_{k} has the property that for any S ⊂ X_{k} either μ(S) = 0 or μ(S) = μ(X_{k})).