A point-process law is determined by its finite-dimensional count laws

Let ${\bar{\mathbb{N}}}_0:={\mathbb{N}}_0\cup \{+\mathrm{\infty }\}$, the range of every count evaluation, and let $\mathcal{C}$ consist of the cylinder sets $\{\mu :\mu (B_j)\in E_j,1\le j\le m\}$ for finite Borel families $B_1,\dots ,B_m$ and subsets $E_j\subseteq {\bar{\mathbb{N}}}_0$. Each such constraint is measurable because this countable range consists of Borel singletons in $[0,+\mathrm{\infty }]$. The family is a pi-system: intersecting two cylinders concatenates their finite coordinate lists, and if a Borel set occurs twice its two constraints are replaced by their intersection. Every evaluation preimage is a one-coordinate cylinder of this form, so by definition of the evaluation sigma-algebra $\sigma (\mathcal{C})$ is the full configuration sigma-algebra.

The hypothesis says the laws $P_{\eta }$ and $P_{{\eta }^{\mathrm{\prime }}}$ assign the same probability to every member of $\mathcal{C}$, since membership in a cylinder is a finite-vector count event. Both are probability measures by [F1]. Applying [F2] to this generating pi-system gives $P_{\eta }=P_{{\eta }^{\mathrm{\prime }}}$.