14.1: Infinite realized causal observable in classical systems

Put $T^2_\le = \{ (t_1,t_2) \in T^2{}: t_1 \le t_2 \}$. An element $t_0 \in T$ is called a root if $t_0 \le t$ ($\forall t \in T$) holds. If $T$ has the root $t_0$, we sometimes denote $T$ by $T(t_0)$. $T' (\subseteq T )$ is called lower bounded if there exists an element $t_i (\in T)$ such that $t_i {{\; \leqq \;}}t $ $(\forall t \in T')$.

Therefore, if $T$ has the root, any $T' (\subseteq T )$ is lower bounded. We always assume that $T$ is complete, that is, for any $T' (\subseteq T )$ which is lower bounded, there exists an element ${\rm Inf}_T ( T' ) (\in T )$ that satisfies the following (i) and (ii):

As typical examples, it suffices to consider the finite tree $T$ (mentioned in $\S$6) such that

or, non-negative real numbers $T(0)=\{ t \in {\mathbb R} \;|\; t {\; \geqq \;}0 \}$ or natual numbers $T(1)=\{1,2,\ldots \}$

Let $(T{(t_0)}, {{\; \leqq \;}})$ be an infinite tree with the root $t_0$. For each $t \in T$, consider the classical basic structure:

Also, for each $t \in T$, define the separable complete metric space $X_t$, and the Borel field $ {\cal B}_{X_t}$, and further, define the observable ${\mathsf O}_t {{=}} (X_t, {\cal F}_t, F_t)$ in ${L^\infty(\Omega_t, \nu_t)}$. That is, we have a sequential causal observable:

Now let us construct the realized causal observable in what follows:

Here, define, $\overline{\cal P}_0(T)$ $( = \overline{\cal P}_0(T(t_0)) \subseteq {\cal P}(T) )$ such that

Let ${T'{(t_0)}} \in \overline{\cal P}_0(T{(t_0)} )$. Since $(T'(t_0), {{\; \leqq \;}})$ is finite, we can put $({T'} {{=}} \{ t_0, t_1,\ldots , t_N \},$ $ \pi{}: {T'} \setminus \{t_0\} \to T')$, where $\pi$ is a parent map.

Review 14.1 [The review of Definition 12.4]
Finite case: The realization of asequential causal observable is easy as follows:

Let $T' (= T'(t_0) ) \in \overline{\mathcal P}_0(T)$. Consider the sequential causal observable $[{}\{ {\mathsf O}_t \}_{ t \in {T'}} , \{ \Phi_{\pi(t), t }{}: $ $L^\infty(\Omega_t, \nu_t) \to L^\infty(\Omega_{\pi(t)}, \nu_{\pi(t)}) \}_{ t \in {T'} \setminus \{t_0 \} }$ $]$. For each $s$ $(\in {T'})$, putting $T_s =\{ t \in T' \;|\; t {\; \geqq \;}s \}$, define the observable $\widehat{\mathsf O}_s {{=}} (\times_{t \in T_s } X_t, $ $\times_{t \in T_s } {\cal F}_t, {\widehat F}_s)$ in $ L^\infty(\Omega_t, \nu_t) $ such that

\begin{align} \widehat{\mathsf O}_s = \left\{\begin{array}{ll} {\mathsf O}_s \quad & \mbox{($ s \in {T'} \setminus \pi ({T'}) $\; \mbox{ and })} \\ \\ {\mathsf O}_s {\times} ({ \underset{{t \in \pi^{-1} (\{ s \})}}{\times} } \Phi_{ \pi(t), t} \widehat {\mathsf O}_t) \quad & \mbox{($ s \in \pi ({T'}) ${}\; \mbox{ and })} \end{array}\right. \tag{14.2} \end{align}

And further, iteratively, we get $\widehat{\mathsf O}_{t_0}{{=}} (\times_{t \in T' } X_t, $ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{t_0})$, which is also denoted by $\widehat{\mathsf O}_{{T'}}{{=}} (\times_{t \in T' } X_t, $ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{T'})$.
$\Big($ In classical cases, the existence is guaranteed by Definition 12.4 $\Big)$

For any subsets $T_{1} \subseteq T_{2}( \subseteq {{T}})$, define the natural map $ \pi_{T_{1},T_{2}}: \times_{{t} \in T_{2}} X_{{t}} \longrightarrow \times_{{t} \in T_{1}} X_{{t}} $ by

It is clear that the observables $\bigl\{$ $\widehat{\mathsf O}_{T'}{{=}} (\times_{t \in T' } X_t, $ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{T'})$ $~|~$ ${T'}\in{\overline{\cal P}_0}({{T}})$ $\bigr\}$ in $L^\infty(\Omega_{t_0}, \nu_{t_0})$ satisfy the following consistency condition , that is,

Then, by Theorem 4.1[ Kolmogorov extension theorem in measurement theory ], there uniquely exists the observable ${\widehat{\mathsf O}}_{{{T}}}$ ${{=}}$ $\bigl(\mathop{\Large{\mbox{ $\times$}}}_{{t} \in {T}}X_{{t}}, $ $\mathop{\mbox{$\boxtimes$}}_{{t}\in{{T}}} {\cal F}_{{t}},$ ${\widehat F}_{{T}} \bigr)$ in $L^\infty(\Omega_{t_0}, \nu_{t_0})$ such that:

This observable ${\widehat{\mathsf O}}_{{{T}}}$ ${{=}}$ $(\mathop{\Large{\mbox{ $\times$}}}_{{t} \in {T}}X_{{t}}, $ $\mathop{\mbox{$\boxtimes$}}_{{t}\in{{T}}} {\cal F}_{{t}},$ ${\widehat F}_{{T}} )$ is called the realization of the sequential causal observable $[{}{\mathbb O}_{T(t_0)}{}]$ $=$ $[{}\{ {\mathsf O}_t \}_{ t \in T} , \{ \Phi_{t_1,t_2}{}: $ $L^\infty(\Omega_{t_2}, \nu_{t_2}) $ $\to L^\infty(\Omega_{t_1}, \nu_{t_1}) \}_{(t_1,t_2) \in T^2_{\leqq}}$ $]$.

Summing up the above argument, we have the following theorem in classical systems. This is the infinite version of Definition 12.4.