A regular Bernoulli submodel and its score

Every subset of $\{0,1\}$ and its complement lie in the full power set, and every countable union of such subsets is still a subset; thus the power set is a sigma-algebra by [F4], and with this sigma-algebra the sample set is a measurable space by [F4]. Counting measure on this two-point set is a measure of total mass two by [F5, F6], hence sigma-finite by the constant sequence of covers $E_n=\{0,1\}$ in [F6]. Its only null set is empty, so each $P_t$ is absolutely continuous with respect to it. The choice of $\delta$ ensures $p+t>0$ and $1-p-t>0$ throughout $\mathrm{\mid }t\mathrm{\mid }<\delta$; the displayed masses sum to one, so each $P_t$ is a probability measure.

Since $p,1-p>0$, [F7] and the definition of derivative give $\sqrt{p+t}=\sqrt{p}+t\mathrm{/}(2\sqrt{p})+o(t)$ and $\sqrt{1-p-t}=\sqrt{1-p}-t\mathrm{/}(2\sqrt{1-p})+o(t)$. These are exactly the two coordinates of $\sqrt{p}+(t\mathrm{/}2)s\sqrt{p}+o(t)$ for the stated $s$; because counting-measure $L^2$ on two points is the Euclidean norm, the combined remainder is $o(t)$. The density versions are explicit, so the calculation uses [A1]'s convention without a selection. Thus the path is DQM with this score by [F1].

Direct calculation gives $Ps=p(1\mathrm{/}p)+(1-p)(-1\mathrm{/}(1-p))=0$ and $P{s}^2=p\mathrm{/}{p}^2+(1-p)\mathrm{/}(1-p{)}^2=1\mathrm{/}[p(1-p)]<\mathrm{\infty }$. Hence the score is in $L_0^2(P)$ and has the stated variance.