Real Analysis Homework #9
Due 12/8 1. Let ν be the Lebesgue measure in [0, 1], and let µ be the counting measure on the same σ-algebra of the Lebesgue measurable subsets of [0, 1]. Show that ν is absolutely continuous with respect to µ. Does there exist a nonnegative µ-measurable function f : [0, 1] → [0, ∞] for which ν(E) = R
Ef dµ for any measurable set E?
2. Let µj, νj be σ-finite measures on (Xj, Bj), j = 1, 2. Assume that νj µj. Then ν1× ν2 µ1× µ2 and
d(ν1× ν2)
d(µ1× µ2)(x1, x2) = dν1
dµ1(x1)dν2 dµ2(x2).
3. Show that the Jordan decomposition is minimal in the sense that if µ is a signed measure and µ = µ1− µ2, where µ1 and µ2 are measures, then
|µ| ≤ µ1+ µ2 with equality only if µ1 = µ+ and µ2 = µ−.
4. Let λ and µ be two measures on (X, B). Suppose that µ is σ-finite and g ≥ 0 measurable. Show that
g = dλ
dµ, i.e., λ(E) = Z
E
gdµ
if and only if for all A ∈ B and α, β ≥ 0,
λ(A ∩ {x : g(x) ≥ α}) ≥ αµ(A ∩ {x : g(x) ≥ α}), λ(A ∩ {x : g(x) < β}) ≤ βµ(A ∩ {x : g(x) < β}).
Hint: For the ”if” part, using these two conditions to show that for µ(A) < ∞ βµ(A ∩ {x : g(x) ∈ [α, β)}) ≥ λ(A ∩ {x : g(x) ∈ [α, β)})
≥ αµ(A ∩ {x : g(x) ∈ [α, β)}).
1
