8.11.E: Problems on Radon-Nikodym Derivatives and Lebesgue Decomposition
Fill in all proof details in Lemma 2 and Theorem 1.
Verify the statement following formula ( 3 ). Also prove the following:
(i) If \(P \in \mathcal{M}\) along with \(-P \in \mathcal{M},\) then \(s \perp t\) implies \(t \perp s\);
(ii) \(s \perp t\) iff \(v_{s} \perp t\).
Prove Corollary 1.
[Hints: Here \(\mathcal{M}\) is a \(\sigma\)-ring. Suppose \(s\) and \(u\) reside in \(P^{\prime}\) and \(P^{\prime \prime},\) respectively, and \(v_{t} P^{\prime}=0=v_{t} P^{\prime \prime} .\) Let \(P=P^{\prime} \cup P^{\prime \prime} \in \mathcal{M} .\) Verify that \(v_{t} P=0\) (use Problem 8 in Chapter \(7, §11 \text { ). Then show that both } s \text { and } u \text { reside in } P .]\)
Show that if \(s: \mathcal{M} \rightarrow E^{*}\) is a signed measure in \(S \in \mathcal{M},\) then \(s^{+} \perp s^{-}\) and \(s^{-} \perp s^{+}\).
Fill in all details in the proof of Theorem \(2 .\) Also prove the following:
(i) \(s^{\prime}\) and \(v_{s^{\prime}}\) are absolutely \(t\)-continuous.
[Hint: Use Theorem 2 in Chapter 7, §11.]
(ii) \(v_{s}=v_{s^{\prime}}+v_{s^{\prime \prime}}, v_{s^{\prime \prime}} \perp t\).
(iii) If \(s\) is a measure \((\geq 0),\) so are \(s^{\prime}\) and \(s^{\prime \prime}\).
Verify Note 3 for Theorem 1 and Corollary \(3 .\) State and prove both generalized propositions precisely.