The Radon-Nikodym Theorem, Part I

In the definition of an additive set function, show that the series $\sum_{n=1}^{\infty } \varphi \left(A_{n}\right)$ must converge absolutely.

In Claim 4, quickly verify that $\varphi^{-}$is a finite measure with support $A^{-}$.

Definition (preceding Claim 4). Let $\left(A^{+}, A^{-}\right)$denote a Hahn decomposition of $\Omega$ with respect to the additive set function $\varphi$. Define $\varphi^{+}: \mathcal{F} \rightarrow \mathbb {R}$ and $\varphi^{-}: \mathcal{F} \rightarrow \mathbb {R}$ by writing $\varphi^{+}(A)=\varphi \left(A \cap A^{+}\right)$and $\varphi^{-}(A)=-\varphi \left(A \cap A^{-}\right)$for every $A \in \mathcal{F}$; $\varphi^{+}$and $\varphi^{-}$are called the upper and lower variation of $\varphi$, and $\varphi =\varphi^{+}-\varphi^{-}$is called the Jordan decomposition of $\varphi$.

Claim 4. Assume the notation above. The functions $\varphi^{+}$and $\varphi^{-}$above are finite measures on $\mathcal{F}$, having supports $A^{+}$and $A^{-}$, respectively. (Therefore, $\varphi^{+}$and $\varphi^{-}$have disjoint supports.) [The proof for $\varphi^{+}$is given in the text; this exercise asks for the analogous argument for $\varphi^{-}$.]

Suppose that ( $A^{+}, A^{-}$) and ( $B^{+}, B^{-}$) are Hahn decompositions with respect to an additive set function $\varphi$. Then $\varphi \left(A^{+} \Delta B^{+}\right)= \varphi \left(A^{-} \Delta B^{-}\right)=0$.

The Jordan decomposition of an additive set function $\varphi$ is unique.

This problem relates somewhat the notion of absolute continuity with the familiar $\varepsilon -\delta$ concepts.

Let $\mu , v, v_{1}$, and $v_{2}$ denote measures, each having common domain $\mathcal{F}$.

Claim 1 may be generalized to the following claim: if $\left\{ E_{n}\right\}_{n=1}^{\infty }$ is a sequence of $\mathcal{F}$-sets such that $\lim E_{n}$ exists [i.e. $\overline{\lim } E_{n}=\underline{\lim } E_{n}$], then $\lim \varphi \left(E_{n}\right)=\varphi \left(\lim E_{n}\right)$.

Claim 1 (Chapter 9). Let $\varphi$ denote an additive set function on $\mathcal{F}$.

Let $f \in L^{1}(\Omega , \mathcal{F}, \mu )$. Define $\nu (E)=\int_{E} f \mathrm{~ d} \mu$ for every $E \in \mathscr {F}$.

Let $\mathfrak {M}$ denote the collection of additive set functions $\varphi$ with domain $\mathcal{F}$.

Let ( $A^{+}, A^{-}$) denote a Hahn decomposition of the additive set function $\varphi$, and let $\varphi =\varphi^{+}-\varphi^{-}$denote the Jordan decomposition. We have

for every $A \in \mathcal{F}$.

The Radon-Nikodym Theorem may fail if $\mu$ is not $\sigma$-finite, even if $v$ is finite.

Let $\mu , v, v_{1}, v_{2}$ and $\rho$ denote $\sigma$-finite measures having domain $\mathcal{F}$. We have the following claims.

Let $\mu$ and $v$ denote finite measures with domain $\mathcal{F}$. Let $\mathcal{H} \subseteq \mathcal{F}$ denote a sub-$\sigma$-field of $\mathcal{F}$. Let $\mu^{*}$and $v^{*}$denote the restrictions of $\mu$ and $v$ to $\mathcal{H}$. In accordance with the Lebesgue Decomposition Theorem, let $v_{\mathrm{ac}}$ and $v_{\mathrm{s}}$ denote the absolutely continuous and singular parts of $v$ with respect to $\mu$. Similarly, let $v_{\mathrm{ac}}^{*}$and $v_{\mathrm{s}}^{*}$denote the absolutely continuous and singular parts of $v^{*}$with respect to $\mu^{*}$. Then

Let $\mu$ and $v$ denote finite measures with domain $\mathcal{F}$. Let $\left\{ v_{n}\right\}_{n=1}^{\infty }$ denote a sequence of finite measures with domain $f$ such that $v=\sum_{n=1}^{\infty } v_{n}$. Assume for each $n \in \mathbb {N}$ and $A \in \mathcal{F}$ that

where each $f_{n}$ is a nonnegative finite-valued function on $\Omega$ and each $v_{n}^{\prime }$is singular with respect to $\mu$. Finally, assume for each $A \in \mathcal{F}$ that

where $f$ is a nonnegative finite-valued function on $\Omega$ and $v^{\prime } \perp \mu$.

Suppose that $\mu$ and $v$ are $\sigma$-finite measures on $\mathcal{F}$. The Lebesgue decomposition of $v$ with respect to $\mu$ is unique. That is, if $v=v_{\mathrm{ac}}+v_{\mathrm{s}}$ where $v_{\mathrm{ac}}$ and $v_{\mathrm{s}}$ are $\sigma$-finite measures with $v_{\mathrm{ac}} \ll \mu$ and $v_{\mathrm{s}} \perp \mu$, and if in addition $\nu =\nu_{\mathrm{ac}}^{\prime }+\nu_{\mathrm{s}}^{\prime }$ where $\nu_{\mathrm{ac}}^{\prime }$ and $\nu_{\mathrm{s}}^{\prime }$ are $\sigma$-finite measures with $\nu_{\mathrm{ac}}^{\prime } \ll \mu$ and $\nu_{\mathrm{s}}^{\prime } \perp \mu$, then $\nu_{\mathrm{ac}}=\nu_{\mathrm{ac}}^{\prime }$ and $\nu_{\mathrm{s}}=\nu_{\mathrm{s}}^{\prime }$.

Let $\mu$ and $v$ denote measures with domain $\mathcal{F}$ and such that $v(E) \leqslant \mu (E)$ for every $E \in \mathcal{F}$. Let $0<p<+\infty$ and $f \in L^{p}(\Omega , \mathcal{F}, \mu )$. Then $\int \left|f\right|^{p} \, \mathrm{d} v \leqslant \int \left|f\right|^{p} \, \mathrm{d} \mu$, so that $f \in L^{p}(\Omega , \mathcal{F}, v)$.

Let $\mu$ and $v$ denote finite measures with common domain $\mathcal{F}$. Let $g \in L^{1}(\Omega , \mathcal{F}, \mu +v)$. Then $g$ is in both $L^{1}(\Omega , \mathcal{F}, \mu )$ and $L^{1}(\Omega , \mathcal{F}, v)$ and

This exercise relies on Exercise 17, which in turn relies on Exercise 16. Let $\mu$ and $v$ denote finite measures with common domain $\mathcal{F}$ and such that $v \ll \mu$. There exists $\mathcal{F}/\mathcal{B}^{*}$-measurable $g$ defined on $\Omega$ with $0 \leqslant g<1$ and such that

Use Exercise 18 [which relies on Exercises 16 and 17] to prove the finite version of the Radon-Nikodym Theorem: Let $\mu$ and $v$ denote finite measures with domain $\mathcal{F}$ and such that $v \ll \mu$. Then there exists a nonnegative finite-valued $\mathcal{F}/\mathcal{B}^{*}$-measurable function $g^{*}$ on $\Omega$ such that $v(E)=\int_{E} g^{*} \, \mathrm{d} \mu$ for every $E \in \mathcal{F}$.

Let $\mu_{0}, \mu_{1}$, and $\mu_{2}$ denote finite measures. Assume that

Then we have for $\left(\mu_{0}+\mu_{1}+\mu_{2}\right)$-a.e. $\omega \in \Omega$ that

Let $\left\{ v_{n}\right\}_{n=1}^{\infty }$ and $\left\{ \mu_{n}\right\}_{n=1}^{\infty }$ denote sequences of finite measures with domain $\mathcal{F}$. For each $n \in \mathbb {N}$ define $\bar{\mu }_{n}=\sum_{i=1}^{n} \mu_{i}$ and $\bar{v}_{n}=\sum_{i=1}^{n} v_{i}$. Define $\mu =\sum_{n=1}^{\infty } \mu_{n}$ and $v=\sum_{n=1}^{\infty } v_{n}$, and further assume that both $\mu$ and $v$ are finite. The goal is to prove the following: If $v_{n} \ll \mu_{n}$ for every $n \in \mathbb {N}$, then $v \ll \mu$ and $\lim_{n} \frac{\mathrm{d} \bar{v}_{n}}{\mathrm{d} \bar{\mu }_{n}}=\frac{\mathrm{d} v}{\mathrm{d} \mu }$ $\mu$-a.e. on $\Omega$. Follow Halmos' outline given below.