
# 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.