Natural and usual augmented Brownian filtrations

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion on a probability space $(\mathrm{\Omega },\mathcal{F},P)$ Brownian motion. First replace the ambient space by its completion, retaining the notation $(\mathrm{\Omega },\mathcal{F},P)$ for this extension. This is supplied by Assuming countable choice, every measure space has a unique complete extension to its completion; AC supplies its countable-choice hypothesis by AC supplies countable selections and prescribed serial paths. The coordinate functions, their laws, and their raw sigma-algebras are unchanged. Four families are distinguished.

1. Raw natural filtration. ${\mathcal{F}}_t^0:=\sigma (B_s:0\le s\le t)$ for $t\ge 0$, and ${\mathcal{F}}_{\mathrm{\infty }}^0:=\sigma \left({\bigcup }_{t\ge 0}{\mathcal{F}}_t^0\right)=\sigma (B_s:s\ge 0)$. Both sigma-algebras exist by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal, and $({\mathcal{F}}_t^0{)}_{t\ge 0}$ is a continuous-time filtration in the sense of Continuous-time filtrations and all-pairs martingales, the smallest one to which $B$ is adapted. It contains no completion and no right-continuous augmentation.
2. Raw right limit. ${\mathcal{F}}_{t+}^0:={\bigcap }_{u>t}{\mathcal{F}}_u^0$ for $t\ge 0$. At $t=0$ this is the germ sigma-algebra used later on this page; the uncountable intersection is the decreasing intersection over the rational $u>t$, since $t\mapsto {\mathcal{F}}_t^0$ is increasing. No null sets are adjoined in this operation.
3. Ambient null ideal and completed raw filtration. Let $\mathcal{N}:=\{M\subseteq \mathrm{\Omega }:\text{there is }{N}_0\in \mathcal{F}\text{ with }P(N_0)=0\text{ and }M\subseteq N_0\}$ be the family of all subsets of ambient $P$-null events. These sets are ambient-measurable because the ambient probability space was completed. This is a sigma-ideal: for a countable family choose null envelopes using AC, then take their union; subsets require the same envelope. Nullness follows from Null sets are closed under countable unions and, in a complete space, under arbitrary subsets. Put $\bar{\mathcal{F}}{}_t^0:=\sigma ({\mathcal{F}}_t^0\cup \mathcal{N}),t\ge 0.$ Each $\bar{\mathcal{F}}{}_t^0$ is a sub-sigma-algebra of $\mathcal{F}$, contains every member of $\mathcal{N}$, and $\bar{\mathcal{F}}{}_s^0\subseteq \bar{\mathcal{F}}{}_t^0$ for $s\le t$, so $(\bar{\mathcal{F}}{}_t^0{)}_{t\ge 0}$ is a filtration.
4. Usual augmentation. ${\mathcal{F}}_t:={\bigcap }_{u>t}\bar{\mathcal{F}}{}_u^0$ for $t\ge 0$.

The following facts are part of the definition and are the form in which it is used later.

(a) ${\mathcal{F}}_t^0\subseteq \bar{\mathcal{F}}{}_t^0\subseteq {\mathcal{F}}_t$ for every $t\ge 0$, and every ${\mathcal{F}}_t$ contains $\mathcal{N}$. (b) $({\mathcal{F}}_t{)}_{t\ge 0}$ is increasing and right-continuous: for every $t\ge 0$, ${\bigcap }_{s>t}{\mathcal{F}}_s={\mathcal{F}}_t\mathrm{.}$ Indeed, if $A\in {\bigcap }_{s>t}{\mathcal{F}}_s$, fix $u>t$ and take $s=(t+u)\mathrm{/}2$. Then $A\in {\mathcal{F}}_s\subseteq \bar{\mathcal{F}}{}_u^0$. Since $u>t$ was arbitrary, $A\in {\mathcal{F}}_t$. Conversely $t\le s$ gives ${\mathcal{F}}_t\subseteq {\mathcal{F}}_s$, because $\bar{\mathcal{F}}{}^0$ is increasing and the intersection defining ${\mathcal{F}}_t$ ranges over the larger parameter set $\{u>t\}\supseteq \{u>s\}$, so ${\mathcal{F}}_t\subseteq {\bigcap }_{s>t}{\mathcal{F}}_s$. (c) Every completed set differs from a raw set by a null set. Let ${\mathcal{D}}_t:=\{A\subseteq \mathrm{\Omega }:\text{there is }{A}_0\in {\mathcal{F}}_t^0\text{ and }M\in \mathcal{N}\text{ with }A\mathrm{△}{A}_0\subseteq M\}\mathrm{.}$ Then ${\mathcal{D}}_t$ is a sigma-algebra containing ${\mathcal{F}}_t^0\cup \mathcal{N}$: complementation preserves symmetric difference, and, after selecting countably many witnesses by AC, $({\bigcup }_k{A}_k)\mathrm{△}({\bigcup }_k{A}_{0,k})\subseteq {\bigcup }_k(A_k\mathrm{△}{A}_{0,k})\subseteq {\bigcup }_k{M}_k\in \mathcal{N}\mathrm{.}$ Thus $\bar{\mathcal{F}}{}_t^0\subseteq {\mathcal{D}}_t$. Conversely, if $A\mathrm{△}{A}_0\subseteq M\in \mathcal{N}$, then $A\mathrm{△}{A}_0\in \mathcal{N}$, so $A=A_0\mathrm{△}(A\mathrm{△}{A}_0)\in \bar{\mathcal{F}}{}_t^0$. This proves equality, and also ambient measurability of every $A\in {\mathcal{D}}_t$. In particular, for $A\in \bar{\mathcal{F}}{}_t^0$ one has $P(A)=P(A_0)$ for such an $A_0$, and ${\int }_{A}XdP={\int }_{A_0}XdP$ for every integrable real or complex $X$, by Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree applied to $X{1}_A$ and $X{1}_{A_0}$. Apply it to indicators for the measure equality. If $P(A)=0$, its raw representative $A_0$ is null, so $A\subseteq A_0\cup M$ is contained in an ambient null envelope. Every subset of $A$ therefore belongs to $\mathcal{N}$. This proves completeness of each $\bar{\mathcal{F}}{}_t^0$. It also proves completeness of ${\mathcal{F}}_t$: a null member belongs to $\bar{\mathcal{F}}{}_{t+1}^0$ and hence all its subsets belong to $\mathcal{N}\subseteq {\mathcal{F}}_t$. Thus every ambient null event and every one of its subsets belongs already to ${\mathcal{F}}_0$; together with right-continuity, this is the usual-conditions convention used below. (d) The four families ${\mathcal{F}}_t^0$, ${\mathcal{F}}_{t+}^0$, $\bar{\mathcal{F}}{}_t^0$ and ${\mathcal{F}}_t$ are kept distinct in every statement below. The strong Markov theorem is stated for $({\mathcal{F}}_t)$ and the deterministic Markov theorems are stated for both $({\mathcal{F}}_t^0)$ and $({\mathcal{F}}_t)$; the companion examples page carries a counterexample showing that ${\mathcal{F}}_{0+}^0\mathrm{\ne }{\mathcal{F}}_0^0$ in the canonical realization.

The raw filtration remains raw even though the ambient measure has been completed; no null set is removed from $\mathrm{\Omega }$. Choice is used for the completion theorem and the countable witnesses above, and is also inherited from the ambient Brownian construction.

Sousi, Definition 6.10 (printed p. 54), defines the natural filtration and its raw right limit; it does not supply the completion construction. Completion is supplied by the declared measure-space theorem, with the ambient-null-ideal convention and the symmetric-difference description proved above. These distinctions are needed when moving between completed events and raw representatives in the later Markov and zero-one arguments.