Blumenthal's zero-one law

Since $A\in {\mathcal{F}}_{0+}^0={\bigcap }_{t>0}{\mathcal{F}}_t^0$, one has $A\in {\mathcal{F}}_t^0$ for every $t>0$, hence $A\in \sigma \left({\bigcup }_{t>0}{\mathcal{F}}_t^0\right)={\mathcal{F}}_{\mathrm{\infty }}^0=\sigma (B_t:t\ge 0)$; equivalently $A=\{B\in A_0\}$ for the corresponding product-measurable set $A_0$ of the path space, that is, $A$ is an event of the sigma-algebra generated by the whole future path $(B_t{)}_{t\ge 0}$. By [F1] we also have $A\in {\mathcal{F}}_0$.

Fix any ${\mathcal{F}}_0$-event $A^{\mathrm{\prime }}$ and any product-measurable path set $\mathrm{\Gamma }\subseteq {\mathbb{R}}^{[0,\mathrm{\infty })}$. Let $j:C([0,\mathrm{\infty }),\mathbb{R})\to {\mathbb{R}}^{[0,\mathrm{\infty })}$ be the measurable inclusion from [F2]. By [F2] at $s=0$ for the bounded Borel functional $1_{\mathrm{\Gamma }}$ of the future path, $E[1_{\mathrm{\Gamma }}((B_t{)}_{t\ge 0})\mathrm{\mid }{\mathcal{F}}_0]={\mathrm{\Psi }}_{1_{\mathrm{\Gamma }}}(B_0)$ almost surely; because $B_0=0$ almost surely and $\mu$ is Wiener measure on the continuous path space, ${\mathrm{\Psi }}_{1_{\mathrm{\Gamma }}}(0)={\int }_C{1}_{\mathrm{\Gamma }}(j(w))\mu (dw)=\mu (j^{-1}(\mathrm{\Gamma }))$. Thus the conditional expectation is the constant $\mu (j^{-1}(\mathrm{\Gamma }))$. The event-integral characterization in [F3] therefore gives $P(A^{\mathrm{\prime }}\cap \{(B_t{)}_{t\ge 0}\in \mathrm{\Gamma }\})=P(A^{\mathrm{\prime }})\mu (j^{-1}(\mathrm{\Gamma }))$; in particular, the product-space future path is independent of the completed time-zero sigma-algebra.

Apply step 1.2 to $A^{\mathrm{\prime }}=A$ and to the path set $\mathrm{\Gamma }:=A_0$ supplied by step 1.1, so that $\{(B_t{)}_{t\ge 0}\in \mathrm{\Gamma }\}=A$. Taking $A^{\mathrm{\prime }}=\mathrm{\Omega }$ in step 1.2 also gives $P(A)=\mu (j^{-1}(\mathrm{\Gamma }))$. Hence $P(A)=P(A\cap A)=P(A)\mu (j^{-1}(\mathrm{\Gamma }))=P(A{)}^2$, so $P(A)\in \{0,1\}$.

Step 2.1 classifies the number $P(A)$, not the set $A$. For example, a Brownian realization may contain a null exceptional outcome at which $B_0\mathrm{\ne }0$; then $\{B_0\mathrm{\ne }0\}\in {\mathcal{F}}_0^0\subseteq {\mathcal{F}}_{0+}^0$ can be nonempty and proper while having probability zero. No right-continuity of the raw filtration at zero is used, only the containment of the germ in the usual time-zero sigma-algebra and the future-path independence at $s=0$. AC is used only through [F4] in the conditional-expectation characterization of step 1.2.