Anytime-valid p-value processes

Let ${\mathcal{P}}_0$ be a family of probability laws on a common discrete filtered probability space $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0})$. A process $p=(p_t{)}_{t\ge 0}$ is an anytime-valid p-value process for ${\mathcal{P}}_0$ if each $p_t$ is ${\mathcal{F}}_t$-measurable with values in $[0,1]$ and, for every $P\in {\mathcal{P}}_0$ and $\alpha \in (0,1)$, $P(\mathrm{\exists }t\ge 0:p_t\le \alpha )\le \alpha \mathrm{.}$

At time $t$, the rejection rule at threshold $\alpha$ is the ${\mathcal{F}}_t$-measurable test function ${\phi }_{t,\alpha }={\mathbf{1}}_{\{p_t\le \alpha \}}$ in the sense of Randomized tests: the indicator supplies a binary probability kernel by Every measurable rejection function defines a randomized test. For any stopping time $\tau :\mathrm{\Omega }\to {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$ Discrete stopping time, the rejection event is $\{\tau <\mathrm{\infty },p_{\tau }\le \alpha \}={\bigcup }_{t\ge 0}(\{\tau =t\}\cap \{p_t\le \alpha \})\mathrm{.}$ For $t=0$, $\{\tau =t\}=\{\tau \le 0\}\in {\mathcal{F}}_0$; for $t\ge 1$, $\{\tau =t\}=\{\tau \le t\}\setminus \{\tau \le t-1\}\in {\mathcal{F}}_t$ by the stopping-time condition and filtration property. Thus each term is in ${\mathcal{F}}_t$, so this event is measurable and is contained in the all-times crossing event; the definition therefore controls rejection after data-dependent stopping.