The ridge estimates and self-normalized sums are measurable

For $s\le t$, each coordinate of $A_s$ is ${\mathcal{F}}_s$-measurable and each coordinate of ${\eta }_s$ is ${\mathcal{F}}_{s+1}$-measurable; the filtration gives ${\mathcal{F}}_s,{\mathcal{F}}_{s+1}\subseteq {\mathcal{F}}_{t+1}$. The reward equation and [F4] make $X_s$ measurable at ${\mathcal{F}}_{t+1}$. Applying [F4] coordinatewise to the finite sums and products defining $G_t$, $S_t$, $V_t$, and $b_t$ proves that each coordinate is ${\mathcal{F}}_{t+1}$-measurable. The Borel sigma-algebra of a finite-dimensional Euclidean vector or matrix space is generated by its coordinate projections: rational open boxes form a countable basis. Thus these maps are measurable random elements. At $t=0$ all four are constant.

The determinant $D_t$ is ${\mathcal{F}}_{t+1}$-measurable by step 1.1, [F4], and the polynomial formula [F5]. By [F8] its range lies in $J=[{\lambda }^d,\mathrm{\infty })$. It is also measurable as a map into $J$: for every relatively open set $J\cap U$, with $U\subseteq \mathbb{R}$ open, $D_t^{-1}(J\cap U)=D_t^{-1}(U)$ because $D_t$ takes values in $J$.

The reciprocal $r:J\to \mathbb{R}$, $r(x)=1\mathrm{/}x$, is continuous since ${\lambda }^d>0$, hence Borel measurable by [F9]. The definition of measurability in [F3] then makes $D_t^{-1}=r\circ D_t$ measurable. By [F5]–[F6], every entry of $\mathrm{adj}(V_t)$ is measurable; for $d=1$ it is the constant $1$. Since $V_t$ is invertible, [F7] gives $\mathrm{adj}(V_t)=D_t{V}_t^{-1}$, so $V_t^{-1}=D_t^{-1}\mathrm{adj}(V_t)$ has measurable entries by [F4].

By [F10], every coordinate of ${\widehat{\theta }}_t=V_t^{-1}{b}_t$ is a finite sum of products of measurable entries from steps 1.1 and 3.1; [F4] makes each coordinate measurable. The coordinate-Borel fact in step 1.1 then makes ${\widehat{\theta }}_t$ a measurable random element. At $t=0$, ${\widehat{\theta }}_0=0$ is constant. Thus every stated vector and matrix is an ${\mathcal{F}}_{t+1}$-measurable random element.