The Hahn-Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon-Nikodym Theorem

Verify the relation $A_{n_{0}} = A+\sum_{j = n_{0}}^{\infty }\left(A_{j}-A_{j+1}\right)$ used in the proof of Lemma 1; namely, if $\left\{ A_{n}\right\} , n \geq 1$, form a nonincreasing sequence and $A = \bigcap_{n = 1}^{\infty } A_{n}$, then for any $n_{0} \geq 1$, the given relation holds. In particular, $A_{1} = \sum_{n = 1}^{\infty }\left(A_{n}-A_{n+1}\right)$ if $\bigcap_{n = 1}^{\infty } A_{n} = \oslash$.

Given the finite measures $\mu_{n}, n = 1,2, \ldots$, on the measurable space $(\Omega , \mathcal{A})$, define a probability measure $\mu$ such that $\mu_{n} \ll \mu$ for all $n$ (i.e., $\mu (A) = 0$ for $A \in \mathcal{A}$ implies $\mu_{n}(A) = 0, n \geq 1$).

Let $U(\alpha , \beta )$ be the uniform distribution over the interval $[\alpha , \beta ](\alpha < \beta )$, and for $n \geq 1$, let $P_{n}$ and $Q_{n}$ be the (probability) measures corresponding to $U\left(-\frac{1}{n}, 1\right)$ and $U\left(0,1+\frac{1}{n}\right)$. Then show that $\left\| P_{n}-Q_{n}\right\| \rightarrow 0$ as $n \rightarrow \infty$, by using part (ii) of Exercise 3.

Let $P$ and $Q$ be two probability measures defined on the measurable space $(\Omega , \mathcal{A})$, and suppose that $P \approx Q$ (i.e., $P \ll Q$ and $Q \ll P$). Let $f = d P / d \mu$ and $g = d Q / d \mu$ for some $\sigma$-finite measure $\mu$ dominating both $P$ and $Q$ (e.g., $\mu = P+Q)$, and let $Z = \log (g / f)$, where as always, $\log$ stands for the natural logarithm. Then (with reference to Exercise 3), show that

Let $P, Q$ and $f, g$ be as in Exercise 3 and, without loss of generality, we suppose that the dominating measure $\mu$ is a probability measure (e.g., $\mu = (P+Q) / 2$). Define $\rho$ by $\rho = \int_{\Omega }(f g)^{1 / 2} d \mu$, and show that:

For $n \geq 1$, let $f_{n}$ be real-valued measurable functions defined on an open set $E \subset \mathbb {R}$ containing 0 , and suppose that $\left|f_{n}(x)\right| \leq M( < \infty )$ for all $n$ and all $x \in E$, and that $f_{n}(x) \underset {n \rightarrow \infty }{\longrightarrow } 0$ for $x \in E$. Then show that, for any sequence $x_{n} \underset {n \rightarrow \infty }{\longrightarrow } 0$, there exists a subsequence $\left\{ x_{m}\right\} \subseteq \left\{ x_{n}\right\}$ and a $\lambda$-null subset $N$ of $E$ (which may depend on $\left\{ x_{n}\right\}$) such that $f_{m}\left(x+x_{m}\right) \underset {m \rightarrow \infty }{\longrightarrow } 0, x \in E-N$; here $\lambda$ is the Lebesgue measure.

For $n \geq 1$, let $f_{n}$ be real-valued measurable functions defined on $\mathbb {R}$, and suppose that $\left|f_{n}(x)\right| \leq M( < \infty )$ for all $n$ and $x \in \mathbb {R}$, and that $\lim_{n \rightarrow \infty } f_{n}^{-}(x) = 0$ for $x \in \mathbb {R}$. Then for any sequence $\left\{ x_{n}\right\}$ with $x_{n} \underset {n \rightarrow \infty }{\longrightarrow } 0$ it holds $\limsup_{n \rightarrow \infty } f_{n}\left(x+x_{n}\right) \geq 0$ a.e. $[\lambda ]$, where $\lambda$ is the Lebesgue measure.