Martingale Theorems

Fill in the details for the outline concerning the proof of (9.12).

Framework (from the section's opening discussion). Let $\Omega$ denote a nonempty set, $\mathcal{F}$ a $\sigma$-field of subsets of $\Omega$, and $\mu$ and $\eta$ arbitrary measures on $\mathcal{F}$. Let $\left\{ \mathcal{F}_{n}\right\}_{n=1}^{\infty }$ be a sequence of sub-$\sigma$-fields of $\mathcal{F}$ with $\bigcup_{n=1}^{\infty } \mathcal{F}_{n} \subseteq \mathcal{F}$. For each $n \in \mathbb {N}$, let $\mu_{n}$ and $\eta_{n}$ denote the restrictions of $\mu$ and $\eta$ to $\mathcal{F}_{n}$, assumed $\sigma$-finite on $\mathcal{F}_{n}$, and let $f_{n}$ denote a derivative of $\eta_{n}$ with respect to $\mu_{n}$. We write $\underline{f}=\underline{\lim }_{n} f_{n}$ and $\bar{f}=\overline{\lim }_{n} f_{n}$, that is, $\underline{f}=\lim_{n} \inf \left\{ f_{n}, f_{n+1}, \cdots \right\}$ and $\bar{f}=\lim_{n} \sup \left\{ f_{n}, f_{n+1}, \cdots \right\}$. In the situation treated by Claim 5, $\mathcal{F}_{1} \subseteq \mathcal{F}_{2} \subseteq \cdots$; write $\mathcal{F}_{\infty }=\sigma \left(\bigcup_{n=1}^{\infty } \mathcal{F}_{n}\right)$, with $\mu_{\infty }$ and $\eta_{\infty }$ denoting the restrictions of $\mu$ and $\eta$ to $\mathcal{F}_{\infty }$.

Claim 5 (restated in part; the two displayed inequalities to be established are numbered (9.11) and (9.12)). Suppose that $\mathcal{F}_{1} \subseteq \mathcal{F}_{2} \subseteq \cdots$. Then $\underline{f}$ and $\bar{f}$ are $\mathcal{F}_{\infty }/\mathcal{B}^{*}$-measurable, and in addition we have the following: (a) $\underline{f}$ and $\bar{f}$ are derivatives of $\eta_{\infty }$ with respect to $\mu_{\infty }$. (b) $\underline{f}=\bar{f}$ $\mu$-a.e. on $\Omega$, so that $\lim_{n} f_{n}$ exists $\mu$-a.e. on $\Omega$. (c) $\int_{A} \lim_{n} f_{n} \, \mathrm{d} \mu =\eta_{\infty , \mathrm{ac}}(A)$ for every $A \in \mathcal{F}_{\infty }$. Establishing (a) reduces (via the sets $L_{\alpha , \underline{f}}=\left\{ \omega \in \Omega : \underline{f}(\omega ) \leqslant \alpha \right\}$ and $G_{\alpha , \bar{f}}=\left\{ \omega \in \Omega : \bar{f}(\omega ) \geqslant \alpha \right\}$, together with Claim 2) to proving

and

This should be straightforward if one understands the details concerning the proof of (9.11).

Prove (1)-(3) and (1')-(3') in Claim 6.

Framework. Assume the same framework as in the preceding exercise, save that here $\mathcal{F}_{1} \supseteq \mathcal{F}_{2} \supseteq \cdots$; write $\mathcal{F}_{0}=\bigcap_{n=1}^{\infty } \mathcal{F}_{n}=\lim_{n} \mathcal{F}_{n}$, with $\mu_{0}$ and $\eta_{0}$ denoting the restrictions of $\mu$ and $\eta$ to $\mathcal{F}_{0}$.

Claim 6 (restated). Suppose that $\mathcal{F}_{1} \supseteq \mathcal{F}_{2} \supseteq \cdots$. Then both $\underline{f}$ and $\bar{f}$ are $\mathcal{F}_{0}/\mathcal{B}^{*}$-measurable, and we have the following: (a) Both $\underline{f}$ and $\bar{f}$ are derivatives of $\eta_{0}$ with respect to $\mu_{0}$. (b) $\underline{f}=\bar{f}$ $\mu$-a.e. on $\Omega$, so that $\lim_{n} f_{n}$ exists $\mu$-a.e. on $\Omega$. (c) $\eta_{0, \mathrm{ac}}(A)=\int_{A} \lim_{n} f_{n} \, \mathrm{d} \mu$ for every $A \in \mathcal{F}_{0}$.

As in the proof of Claim 5, establishing part (a) reduces to proving, for every $\alpha \in \mathbb {R}$, the two displays

and

where $M_{\alpha }=\left\{ \omega \in \Omega : \inf \left\{ f_{1}(\omega ), f_{2}(\omega ), \cdots \right\} <\alpha \right\}$ and $H_{\alpha }=\left\{ \omega \in \Omega : \sup \left\{ f_{1}(\omega ), f_{2}(\omega ), \cdots \right\} >\alpha \right\}$.

To prove (9.18), fix $\alpha \in \mathbb {R}$ and, for each $n \in \mathbb {N}$, define

and $J_{n, n}=\left\{ \omega \in \Omega : f_{n}(\omega )<\alpha \right\}$. The claims to be proved are:

1. $J_{n, p} \in \mathcal{F}_{p}$ for every $n \in \mathbb {N}$ and $p \in \left\{ 1, \cdots , n\right\}$.

2. For each $n \in \mathbb {N}$ the finite sequence $\left\{ J_{n, p}\right\}_{p=1}^{n}$ is disjoint with union $J_{n}$.

3. $J_{1} \subseteq J_{2} \subseteq \cdots$ and $\lim_{n} J_{n}=\bigcup_{n=1}^{\infty } J_{n}=M_{\alpha }$. To prove (9.19), fix $\alpha \in \mathbb {R}$ and, for each $n \in \mathbb {N}$, define

and $J_{n, n}=\left\{ \omega \in \Omega : f_{n}(\omega )>\alpha \right\}$. The claims to be proved are:

1. For every $n \in \mathbb {N}$ and $p=1, \cdots , n$ we have $J_{n, p} \in \mathcal{F}_{p}$.

2. For each $n \in \mathbb {N}$ the finite sequence $\left\{ J_{n, p}\right\}_{p=1}^{n}$ is disjoint with union $J_{n}$.

3. $J_{1} \subseteq J_{2} \subseteq \cdots$ and $\lim_{n} J_{n}=\bigcup_{n=1}^{\infty } J_{n}=H_{\alpha }$.

Assume the framework of Claim 5. If $\eta_{n} \ll \mu_{n}$ for every $n \in \mathbb {N}$, and if for every $E$ in the field $\bigcup_{k=1}^{\infty } \mathcal{F}_{k}$ we have $\lim_{n} \int_{E} f_{n} \, \mathrm{d} \mu =\int_{E} f_{\infty } \, \mathrm{d} \mu$, then $\eta_{\infty } \ll \mu_{\infty }$. Here $f_{\infty }$ denotes a derivative of $\eta_{\infty }$ with respect to $\mu_{\infty }$.

Framework of Claim 5 (restated). Let $\Omega$ denote a nonempty set, $\mathcal{F}$ a $\sigma$-field of subsets of $\Omega$, and $\mu$ and $\eta$ arbitrary measures on $\mathcal{F}$. Let $\mathcal{F}_{1} \subseteq \mathcal{F}_{2} \subseteq \cdots$ be a nondecreasing sequence of sub-$\sigma$-fields of $\mathcal{F}$. For each $n \in \mathbb {N}$, let $\mu_{n}$ and $\eta_{n}$ denote the restrictions of $\mu$ and $\eta$ to $\mathcal{F}_{n}$, assumed $\sigma$-finite on $\mathcal{F}_{n}$, and let $f_{n}$ denote a derivative of $\eta_{n}$ with respect to $\mu_{n}$. Write $\mathcal{F}_{\infty }=\sigma \left(\bigcup_{n=1}^{\infty } \mathcal{F}_{n}\right)$, with $\mu_{\infty }$ and $\eta_{\infty }$ denoting the restrictions of $\mu$ and $\eta$ to $\mathcal{F}_{\infty }$.