The Radon-Nikodym Theorem

There are two ways to show that the convergence in (32.1) must be absolute: Use the Jordan decomposition. Use the fact that a series converges absolutely if it has the same sum no matter what order the terms are taken in.

If $A^{+} \cup A^{-}$is a Hahn decomposition of $\varphi$, there may be other ones $A_{1}^{+} \cup A_{1}^{-}$. Construct an example of this. Show that there is uniqueness to the extent that $\varphi \left(A^{+} \Delta A_{1}^{+}\right)=\varphi \left(A^{-} \Delta A_{1}^{-}\right)=0$.

Show that absolute continuity does not imply the $\epsilon -\delta$ condition (32.4) if $\nu$ is infinite. Hint. Let $\mathscr {F}$ consist of all subsets of the space of integers, let $\nu$ be counting measure, and let $\mu$ have mass $n^{-2}$ at $n$. Note that $\mu$ is finite and $\nu$ is $\sigma$-finite.

Show that the Radon-Nikodym theorem fails if $\mu$ is not $\sigma$-finite, even if $\nu$ is finite. Hint: Let $\mathscr {F}$ consist of the countable and the cocountable sets in an uncountable $\Omega$, let $\mu$ be counting measure, and let $\nu (A)$ be 0 or 1 as $A$ is countable or cocountable.

Let $\mu$ be the restriction of planar Lebesgue measure $\lambda_{2}$ to the $\sigma$-field $\mathscr {F}=\left\{ A \times R^{1}: A \in \mathscr {R}^{1}\right\}$ of vertical strips. Define $\nu$ on $\mathscr {F}$ by $\nu \left(A \times R^{1}\right)=\lambda_{2}(A \times (0,1))$. Show that $\nu$ is absolutely continuous with respect to $\mu$ but has no density. Why does this not contradict the Radon-Nikodym theorem?

Let $\mu , \nu$, and $\rho$ be $\sigma$-finite measures on $(\Omega , \mathscr {F})$. Assume the Radon-Nikodym derivatives here are everywhere nonnegative and finite.

Show that there is a Lebesgue decomposition (32.8) in the $\sigma$-finite as well as the finite case. Prove that it is unique.

The Radon-Nikodym theorem holds if $\mu$ is $\sigma$-finite, even if $\nu$ is not. Assume at first that $\mu$ is finite (and $\nu \ll \mu$ ).

Let $\mu$ and $\nu$ be finite measures on $(\Omega , \mathscr {F})$, and suppose that $\mathscr {F}^{\circ }$ is a $\sigma$-field contained in $\mathscr {F}$. Then the restrictions $\mu^{\circ }$ and $\nu^{\circ }$ of $\mu$ and $\nu$ to $\mathscr {F}^{\circ }$ are measures on $(\Omega , \mathscr {F}^{\circ })$. Let $\nu_{\mathrm{ac}}, \nu_{\mathrm{s}}, \nu_{\mathrm{ac}}^{\circ }, \nu_{\mathrm{s}}^{\circ }$ be, respectively, the absolutely continuous and singular parts of $\nu$ and $\nu^{\circ }$ with respect to $\mu$ and $\mu^{\circ }$. Show that $\nu_{\mathrm{ac}}^{\circ }(E) \geq \nu_{\mathrm{ac}}(E)$ and $\nu_{\mathrm{s}}^{\circ }(E) \leq \nu_{\mathrm{s}}(E)$ for $E \in \mathscr {F}^{\circ }$.

Suppose that $\mu , \nu , \nu_{n}$ are finite measures on $(\Omega , \mathscr {F})$ and that $\nu (A)=\sum_{n} \nu_{n}(A)$ for all $A$. Let $\nu_{n}(A)=\int_{A} f_{n} d \mu +\nu_{n}^{\prime }(A)$ and $\nu (A)=\int_{A} f d \mu +\nu^{\prime }(A)$ be the decompositions (32.8); here $\nu^{\prime }$ and $\nu_{n}^{\prime }$ are singular with respect to $\mu$. Show that $f=\sum_{n} f_{n}$ except on a set of $\mu$-measure 0 and that $\nu^{\prime }(A)=\sum_{n} \nu_{n}^{\prime }(A)$ for all $A$. Show that $\nu \ll \mu$ if and only if $\nu_{n} \ll \mu$ for all $n$.

32.2 ↑ Absolute continuity of a set function $\varphi$ with respect to a measure $\mu$ is defined just as if $\varphi$ were itself a measure: $\mu (A)=0$ must imply that $\varphi (A)=0$. Show that, if this holds and $\mu$ is $\sigma$-finite, then $\varphi (A)=\int_{A} f d \mu$ for some integrable $f$. Show that $A^{+}=[\omega : f(\omega ) \geq 0]$ and $A^{-}=[\omega : f(\omega )<0]$ give a Hahn decomposition for $\varphi$. Show that the three variations satisfy $\varphi^{+}(A)= \int_{A} f^{+} d \mu , \varphi^{-}(A)=\int_{A} f^{-} d \mu$, and $\left|\varphi \right|(A)=\int_{A}\left|f\right| d \mu$. Hint: To construct $f$, start with (32.2).

$\uparrow$ A signed measure $\varphi$ is a set function that satisfies (32.1) if $A_{1}, A_{2}, \ldots$ are disjoint and may assume one of the values $+\infty$ and $-\infty$ but not both. Extend the Hahn and Jordan decompositions to signed measures

$31.22 \uparrow$ Suppose that $\mu$ and $\nu$ are a probability measure and a $\sigma$-finite measure on the line and that $\nu \ll \mu$. Show that the Radon-Nikodym derivative $f$ satisfies

on a set of $\mu$-measure 1.

Find on the unit interval uncountably many probability measures $\mu_{p}, 0<p<1$, with supports $S_{p}$ such that $\mu_{p}\left\{ x\right\} =0$ for each $x$ and $p$ and the $S_{p}$ are disjoint in pairs.

Let $\mathscr {F}_{0}$ be the field consisting of the finite and the cofinite sets in an uncountable $\Omega$. Define $\varphi$ on $\mathscr {F}_{0}$ by taking $\varphi (A)$ to be the number of points in $A$ if $A$ is finite, and the negative of the number of points in $A^{c}$ if $A$ is cofinite. Show that (32.1) holds (this is not true if $\Omega$ is countable). Show that there are no negative sets for $\varphi$ (except the empty set), that there is no Hahn decomposition, and that $\varphi$ does not have bounded range.