8.11.E: Problems on Radon-Nikodym Derivatives and Lebesgue Decomposition

Exercise $\PageIndex{1}$

Fill in all proof details in Lemma 2 and Theorem 1.

Exercise $\PageIndex{2}$

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 $v_{s} \perp t$.

Exercise $\PageIndex{3}$

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 $7, §11 \text { ). Then show that both } s \text { and } u \text { reside in } P .]$

Exercise $\PageIndex{4}$

Show that if $s: \mathcal{M} \rightarrow E^{*}$ is a signed measure in $S \in \mathcal{M},$ then $s^{+} \perp s^{-}$ and $s^{-} \perp s^{+}$.

Exercise $\PageIndex{5}$

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 $s^{\prime \prime}$.

Exercise $\PageIndex{6}$

Verify Note 3 for Theorem 1 and Corollary $3 .$ State and prove both generalized propositions precisely.