Let \(X : T \times \Omega \to E\) be a predictable process, we will show that it is progressively measurable. Namely, fixing \(t \in T\), denoting

we need to show that \(\iota _t \circ X : [0, t] \times \Omega \to E\) is measurable with respect to \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\).

Denoting \(\Sigma _{\mathcal{F}}\) for the predictable \(\sigma \)-algebra generated by \(\mathcal{F}\), as \(u\) is predictable, we have that \(X^{-1}(\mathcal{B}(E)) \le \Sigma _{\mathcal{F}}\). Thus, to show that \(\iota _t \circ X\) is \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable, it suffices to show that \(\iota _t^{-1}(\Sigma _{\mathcal{F}}) \le \mathcal{B}([0, t]) \otimes \mathcal{F}_t\). In particular, as

is suffices to show that sets of the form \(\iota _t^{-1}((s, \infty ) \times A)\) for some \(s \in T, A \in \mathcal{F}_s\) and \(\iota _t^{-1}(\{ \bot \} \times A)\) for some \(A \in \mathcal{F}_\bot \) are \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable.

Indeed, if \(A \in \mathcal{F}_\bot \)

while for any \(s \in T\) and \(A \in \mathcal{F}_s\),

By the monotonicity of the filtration \(\mathcal{F}\), all of these cases are \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable allowing us to conclude.

For \(t \le s\), the set in question is empty and thusly, trivially measurable. On the other hand, for \(s {\lt} t\), measurability follows as \((s, t] \times A = (s, \infty ) \times A \setminus (t, \infty ) \times A\).

Suppose first that \(X\) is predictable. Straightaway, \(X_0\) is \(\mathcal{F}_0\)-measurable as predictable implies progressively measurable which in turn implies adapted.

Fixing \(n\), we observe that for any \(S \in \mathcal{B}(E)\),

where

and

As \(X^{-1}(S) \in \Sigma _{\mathcal{F}}\) – the predictable \(\sigma \)-algebra, it suffices to show that \(\pi ^{-1}(\iota ^{-1}(\Sigma _{\mathcal{F}})) \in \mathcal{F}_n\). To this end, we again only need to show these for the generating sets of \(\Sigma _{\mathcal{F}}\):

• For \(A \in \mathcal{F}_0\), measurability is clear as \(\iota ^{-1}(\{ 0\} \times A) = \varnothing \).

• Similarly, for \(m {\gt} n\) and \(A \in \mathcal{F}_m\), \(\iota ^{-1}((m, \infty ) \times A) = \varnothing \).

• For \(m \le n\) and \(A \in \mathcal{F}_m \le \mathcal{F}_n\) we have that \(\pi ^{-1}(\iota ^{-1}((m, \infty ) \times A)) = A\) which is \(\mathcal{F}_n\) measurable by the monotonicity of the filtration.

Now, supposing \(X_0\) is \(\mathcal{F}_0\)-measurable and \(X_{n + 1}\) is \(\mathcal{F}_n\)-measurable, we will show that \(X\) is predictable. Indeed, fixing \(S \in \mathcal{B}(E)\), we have

Thus, as \(\{ 0\} \times X_0^{-1}(S) \in \Sigma _{\mathcal{F}}\) by construction and \(\{ n + 1\} \times X_{n + 1}^{-1}(S) = (n, n + 1] \times X_{n + 1}^{-1}(S) \in \Sigma _{\mathcal{F}}\) by Lemma 7.13 and the fact that \(X_{n + 1}^{-1}(S) \in \mathcal{F}_n\), we have that \(X^{-1}(S) \in \Sigma _{\mathcal{F}}\) as required.

Take \(\tau _n = \infty \) for all \(n\).

Let \(X \in P_{\mathrm{loc}}\). Then there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0} \in P\). Since \(P \subseteq Q\), for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0} \in Q\). Thus \(X \in Q_{\mathrm{loc}}\).

Let \(X\) be a process in \((P_{\mathrm{loc}})_{\mathrm{loc}}\). By definition there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is in \(P_{\mathrm{loc}}\). By definition of \(P_{\mathrm{loc}}\), for each \(n\) there exists a localizing sequence \((\sigma _{n,k})_{k \in \mathbb {N}}\) such that for all \(k \in \mathbb {N}\), \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k}}\mathbb {I}_{\sigma _{n,k} {\gt} 0}\) is in \(P\).

For each \(n\), since \(\sigma _{n,k} \to \infty \) a.s. as \(k \to \infty \), we may choose \(k_n \in \mathbb {N}\) such that \(P(\sigma _{n,k_n} {\lt} \tau _n \wedge n) \le 2^{-n}\). Let \(\tau '_n = \tau _n \wedge \sigma _{n,k_n}\). \(\tau _n' \to \infty \) by the Borel-Cantelli lemma. Let \(\tau ''_n = \inf _{m \ge n} \tau '_m\). Then \((\tau ''_n)_{n \in \mathbb {N}}\) is a localizing sequence.

It remains to argue that by stability of \(P\), \(X^{\tau ''_n}\mathbb {I}_{\tau ''_n {\gt} 0}\) is in \(P\) for all \(n\). Indeed, \(X^{\tau ''_n} = ((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0})^{\tau ''_n}\mathbb {I}_{\tau ''_n {\gt} 0}\). \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0}\) is in \(P\) by construction and \(P\) is stable.

Since \(X \in P_{\mathrm{loc}}\), then \(X \in (Q_{\mathrm{loc}})_{\mathrm{loc}}\) by assumption and Lemma 7.22. By Lemma 7.23, \((Q_{\mathrm{loc}})_{\mathrm{loc}} = Q_{\mathrm{loc}}\). Thus \(X \in Q_{\mathrm{loc}}\).

This follows from Lemma 7.19.