Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment XII.

1. Problem 1. a) Let μ be a complex measure on a σ-algebra Σ, and let E ∈ Σ. Define λ(E) := sup ∑ |μ(E_{i})|, the supremum being taken over all finite partitions {E_{i}} of E. Does it follow that λ = |μ|?

b) Define a measure λ on Lebesgue measurable subsets of R^{2} as follows

λ(A) := ∫_{A∩R} f(x, 0)dx, A ⊂ R^{2},

Here f(x, 0) is a continuous function having a compact support on R = {(x, y) : y = 0}. Prove that λ is absolutely continuous with respect to the Lebesgue measure on R, but is singular with respect to the Lebesgue measure on R^{2}.

2. Problem 2.

Definition. We say that two measures λ_{1}, λ_{2}, defined on a σ-algebra Σ are mutually singular, (λ_{1} ⊥ λ_{2}), if there exists a pair of disjoint sets A, B such that λ_{1} is concentrated on A, and λ_{2} is concentrated on B. Suppose that μ, λ, λ_{1}, λ_{2} are measures on a σ-algebra Σ, and μ is positive. Prove the following chain of stements.

a) If λ is concentrated on A, so is |λ|.

b) If λ_{1} ⊥ λ_{2}, then |λ_{1}| ⊥ |λ_{2}|.

c) If λ_{1} ⊥ μ, and λ_{2} ⊥ μ, then (λ_{1} + λ_{2}) ⊥ μ.

d) If λ_{1} ≪ μ and λ_{2} ≪ μ, then (λ_{1} + λ_{2}) ≪ μ.

e) If λ ≪ μ, then |λ| ≪ μ.

f) If λ_{1} ≪ μ and λ_{2} ⊥ μ, then λ_{1} ⊥ λ_{2}.

g) If λ ≪ μ and λ ⊥ μ, then λ = 0.

3. Problem 3. Suppose μ and λ are measures on a σ-algebra Σ, μ is positive, λ is complex. Prove that the following two conditions are equivalent: ℵ) λ ≪ μ. ℶ) For every ϵ > 0 there exists a δ > 0 such that |λ(E)| < ϵ for all E ∈ Σ with μ(E) < δ. Hint. To show that ℵ) implies ℶ), assume that ℶ) is false. Then there exists ϵ > 0 and E_{n} ∈ Σ such that μ(E_{n}) < 2^{−n}, but |λ(E_{n})| ≥ ϵ. Put A_{n} := ∪_{i=n}^{∞}E_{i}, A := ∪_{n=1}^{∞}A_{n}. Prove that μ(A) = 0, but |λ|(A) > 0. Use Problem 2, e), to get a contradiction.

4. Problem 4. Suppose μ is a positive measure on Σ, g ∈ L(X, μ), and λ(E) = ∫_{E} fdμ,

E ∈ Σ. Prove that |λ|(E) = ∫_{E} |f|dμ.

Hint. Observe that |λ|(E) ≤ ∫_{E} |f|dμ. To show the opposite inequality, construct

a sequence (g_{n}(x))_{n=1}^{∞} of measurable simple functions such that |g_{n}(x)| = 1, and lim_{n→∞} g_{n}(x)f(x) = |f(x)|. Check that

| ∫_{A} g_{n}fdμ| = | ∑_{j} a_{n,j} ∫_{A∩A_{n,j}} fdμ| = | ∑_{j} a_{n,j}λ(A ∩ A_{n,j})| ≤ ∑_{j} |λ(A ∩ A_{n,j})| ≤ |λ|(A),

where a_{n,j} are the values of g_{n}, attained on the sets A_{n,j}.

5. Problem 5. Let (r_{n})_{n=1}^{∞} be an enumeration of the rational numbers, and for each positive integer n, let f_{n} : R → R be defined as f_{n}(x) = 2^{n+1}, x ∈ [r_{n} − 2^{−n}, r_{n} + 2^{−n}], and zero otherwise. Define a measure λ on Borel subsets of R by

λ(A) := ∫_{A} f(x)dx, f(x) := ∑_{n=1}^{∞} f_{n}(x).

a) Show that f(x) is finite almost everywhere with respect to m, the Lebesgue measure on R.

Hint. Define A_{k} := {x ∈ R : f(x) ≥ 2^{k}}, k is a nonnegative integer. To show that

m(∩_{k=1}^{∞}A_{k}) = 0 observe that ∑_{k=1}^{∞} m(A_{k}) < ∞.

b) Show that λ is σ-finite. In other words find a partition of R into disjoint sets B_{k}, such that R = ∪_{k=1}^{∞}B_{k}, and λ(B_{k}) < ∞.

Hint. One can take

B_{1} := A_{1}^{c}, B_{k} := A_{k}^{c} \ (∪_{l=1}^{k−1}A_{l}^{c}).

c) Show that λ ≪ m.

d) Show that each non-empty open subset of R has infinite measure under λ.

Hint. Observe that f(x) ≥ ∑_{k=1}^{∞} f_{n_{k}}(x), where a subsequence n_{k} is chosen such that

segments [r_{n_{k}} − 2^{−n_{k}}, r_{n_{k}} + 2^{−n_{k}}], [r_{n_{l}} − 2^{−n_{l}}, r_{n_{l}} + 2^{−n_{l}}] are disjoint (for k ≠ l) and belong to the given open interval. Use ∫_{R} f_{n}(x)dx = 1.