Lemma: Witness determination

Agreement of two distributions on $\mathcal{H}$ determines their equality exactly when ${\mathcal{H}}^{\perp }=\{0\}$. In particular, density of $\mathcal{H}$ in the full test space is sufficient.

Proof. The difference of the distributions belongs to ${\mathcal{H}}^{\perp }$. This proves the first assertion. If $\mathcal{H}$ is dense, continuity makes that difference vanish on every test.