Densities and derivatives

Let $\mu$ be the measure defined for all subsets of $X$ by $\mu A = +\infty$ if $A \neq \emptyset$, and $\mu (\emptyset ) = 0$. Show that $\mu f = \mu (2 f)$ for all nonnegative functions $f$.

Let $\lambda$ denote the measure on $\mathbb {R}$ with $\lambda A$ equal to the number of points in $A$ (possibly infinite), and let $\mu A = \infty$ for all nonempty $A$. Show that $\lambda$ has no density with respect to $\mu$, even though both measures have the same negligible sets.

Suppose $\Delta_{1}$ and $\Delta_{2}$ are functions in $\mathcal{M}^{+}(\mathcal{X}, \mathcal{A})$ for which $\mu \left(f \Delta_{1}\right) = \mu \left(f \Delta_{2}\right)$ for all $f$ in $\mathcal{M}^{+}(X, \mathcal{A})$, for some measure $\mu$.

Let $P$ and $Q$ be probability measures on ($\Omega , \mathcal{F}$), and let $\mathcal{E}$ be a generating class for $\mathcal{F}$ that is stable under finite intersections and contains $\Omega$. Suppose there exists a $p$ in $\mathcal{L}^{1}(Q)$ such that $P E = Q(p E)$ for every $E$ in $\mathcal{E}$. Show that $P$ has density $p$ with respect to $Q$.

Does Example $\langle 3 \rangle$ have an analog for $\mathcal{L}^{p}$ convergence, for some $p > 1$ : if $v$ is a finite measure dominated by a measure $\lambda$, and if $\left\{ f_{n}\right\}$ is a sequence of measurable functions for which $\lambda \left|f_{n}\right|^{p} \rightarrow 0$, does it follow that $\nu \left|f_{n}\right|^{p} \rightarrow 0$ ?

Let $\nu$ and $\mu$ be finite measures on the sigma-field $\sigma (\mathcal{E})$ generated by a countable field $\mathcal{E}$. Suppose that for each $\epsilon > 0$ there exists a $\delta > 0$ such that $\nu E < \epsilon$ for each $E$ in $\mathcal{E}$ with $\mu E < \delta$. Show that $v$ is absolutely continuous with respect to $\mu$, as measures on $\sigma (\mathcal{E})$.

Let $v$ and $\mu$ be finite measures on $(X, \mathcal{A})$. Suppose $v$ has Lebesgue decomposition $(\Delta , \mathcal{N})$ with respect to $\mu$. Find the Lebesgue decomposition of $\mu$ with respect to $v$.

Show that a measure $\mu$ is sigma-finite if and only if there exists a strictly positive, measurable function $\Psi$ for which $\mu \Psi < \infty$.

Suppose $\left(\mathcal{N}_{1}, \Delta_{1}\right)$ and $\left(\mathcal{N}_{2}, \Delta_{2}\right)$ are both Lebesgue decompositions for a finite measure $v$ with respect to a finite measure $\mu$. Prove $\mathcal{N}_{1} = \mathcal{N}_{2}$ a.e. $[\nu +\mu ]$ and $\Delta_{1} \mathcal{N}_{1}^{c} = \Delta_{2} \mathcal{N}_{2}^{c}$ a.e. $[\nu +\mu ]$ by the following steps.

Suppose $\nu$ and $\mu$ are both sigma-finite measures on $(X, \mathcal{A})$.

Let $\nu$ be a finite measure for which $\nu = \nu_{1}+\lambda_{1} = \nu_{2}+\lambda_{2}$, where each $\nu_{i}$ is dominated by a fixed sigma-finite measure $\mu$ and each $\lambda_{i}$ is singular with respect to $\mu$. Show that $\nu_{1} = \nu_{2}$ and $\lambda_{1} = \lambda_{2}$.

Let $\mu_{1}$ and $\mu_{2}$ be finite measures with densities $m_{1}$ and $m_{2}$ with respect to a sigma-finite measure $\lambda$. Define $\left(\mu_{1}-\mu_{2}\right)^{+}$ as the measure with density $\left(m_{1}-m_{2}\right)^{+}$ with respect to $\lambda$.

Let $P$ and $Q$ be probability measures with densities $p$ and $q$ with respect to a sigma-finite measure $\lambda$. For fixed $\alpha \geq 1$, show that $\Delta_{\alpha }(P, Q): = \lambda \left|p^{1 / \alpha }-q^{1 / \alpha }\right|^{\alpha }$ does not depend on the choice of dominating measure.

Adapt the argument from the previous Problem to show that the relative entropy $D(P \| Q)$ does not depend on the choice of dominating measure.

Let $P$ be the standard Cauchy distribution on the real line, and let $Q$ be the standard normal distribution. Show that $D(P \| Q) = \infty$, even though $P$ and $Q$ are mutually absolutely continuous.

Let $P$ and $Q$ be probability measures defined on the same sigma-field $\mathcal{F}$. A randomized test is a measurable function $f$ with $0 \leq f \leq 1$. (For observation $\omega$, the value $f(\omega )$ gives the probability of rejecting the "hypothesis" $P$.) Find the test that minimizes $P f+Q(1-f)$.

Let $P$ and $Q$ be finite measures defined on the same sigma-field $\mathcal{F}$, with densities $p$ and $q$ with respect to a measure $\mu$. Suppose $X_{0}$ is a measurable set with the property that there exists a nonnegative constant $K$ such that $q \geq K p$ on $X_{0}$ and $q \leq K p$ on $X_{0}^{c}$. For each $\mathcal{F}$-measurable function with $0 \leq f \leq 1$ and $P f \leq P X_{0}$, prove that $Q f \leq Q X_{0}$.

Let $\mu = \mu_{1}-\mu_{2}$ and $\mu^{\prime } = \mu_{1}^{\prime }-\mu_{2}^{\prime }$ be signed measures, in the sense of Section 3. Show that $v: = \left(\mu_{1}+\mu_{2}^{\prime }\right) \wedge \left(\mu_{1}^{\prime }+\mu_{2}\right)-\left(\mu_{2}+\mu_{2}^{\prime }\right)$ is the largest signed measure for which $v \leq \mu$ and $v \leq \mu^{\prime }$.

Let $f(x) = (1+x) \log (1+x)-x$, for $x \geq -1$. Use the representations

to show that $f(x) \geq 1 / 2 x^{2}(1+x / 3)^{-1}$.

Show that the function $x^{-1} \cos \left(x^{-2}\right)$ is not integrable on [0,1].

Prove Theorem $\langle 19 \rangle$ by establishing the following assertions.

Theorem 19: Let $H$ be a continuous function defined on an interval $[a, b]$, with a (real-valued) right-hand derivative $h(x): = \lim_{\delta \downarrow 0}\left(H(x+\delta )-H(x)\right) / \delta$ existing at each point of $[a, b)$. If $h$ is Lebesgue integrable then $H(x) = H(a)+\int_{a}^{x} h(t) d t$ for each $x$ in $[a, b]$, and hence $H$ is an absolutely continuous function on $[a, b]$.

With no loss of generality, assume $a = 0$ and $b = 1$ and $H(0) = 0$. Write $m$ for Lebesgue measure on $\mathcal{B}[0,1]$. Fix $\epsilon > 0$. Define $A_{n}: = \left\{ n \epsilon \leq h < (n+1) \epsilon \right\}$ for each integer $n$, both positive and negative. Note that

Let $H$ be a continuous real function defined on [0,1). Suppose the right-hand derivative $h$ exists and is finite everywhere. If $h$ is an increasing function, show that $H$ is convex.

Let $h$ be integrable with respect to Lebesgue measure $\mathfrak {m}$ on $\mathbb {R}^{d}$. Show that

Generalize by replacing open balls by decreasing sequences of closed sets $F_{n} \downarrow \left\{ z\right\}$ that are regular in the sense of Section 5.