A test supermartingale gives anytime-valid p-values

The map $g:\mathbb{R}\to [0,1]$ defined by $g(x)=1$ for $x\le 0$ and $g(x)=\min (1,1\mathrm{/}x)$ for $x>0$ is Borel measurable. Thus $p_t=g(E_t)$ is ${\mathcal{F}}_t$-measurable and $[0,1]$-valued. For $0<\alpha <1$, the threshold $1\mathrm{/}\alpha >1$, so $p_t\le \alpha$ holds exactly when $E_t\ge 1\mathrm{/}\alpha$.

Taking the union over all $t$ in the equivalence of step 1.1 and applying F3 gives $P(\mathrm{\exists }t\ge 0:p_t\le \alpha )=P(\mathrm{\exists }t\ge 0:E_t\ge 1\mathrm{/}\alpha )\le P({\sup }_{t\ge 0}{E}_t\ge 1\mathrm{/}\alpha )\le \alpha$ for every $P\in {\mathcal{P}}_0$. This is the defining anytime-valid condition.

For a fixed $t$ and $0<u<1$, $\{p_t\le u\}\subseteq \{\mathrm{\exists }s:p_s\le u\}$, so its probability is at most $u$ by step 2.1. Since $E_t$ is finite, $p_t>0$; hence the bound at $u=0$ is zero, and at $u=1$ it is trivial. For any stopping time $\tau$, $\{\tau <\mathrm{\infty },p_{\tau }\le u\}={\bigcup }_{t\ge 0}(\{\tau =t\}\cap \{p_t\le u\})$ is contained in the all-times crossing event for $0<u<1$; the endpoints follow as above. Thus fixed-time and stopping-time validity both hold.

If ${\mathcal{P}}_0$ is nonempty, form the fixed-time experiment with data space $(\mathrm{\Omega },{\mathcal{F}}_t)$, parameter set $\mathrm{\Theta }={\mathcal{P}}_0$, laws given by restriction to ${\mathcal{F}}_t$, and null set ${\mathrm{\Theta }}_0=\mathrm{\Theta }$. The observation is the identity map from the original sample space to this data space. Step 1.1 makes $p_t$ a statistic and step 3.1 gives its superuniform inequality under every law in ${\mathrm{\Theta }}_0$, so F4 makes it a valid p-value in this experiment. If ${\mathcal{P}}_0$ is empty, all lawwise assertions are vacuous.