Mean measures and measurable nonnegative integrals of random measures

The function $\nu (B)=\mathbb{E}[\zeta (B)]$ is defined for every Borel set $B$ by [F1] and [F2], and $\nu (\mathrm{\varnothing })=0$ because each measure ${\zeta }_{\omega }$ vanishes at $\mathrm{\varnothing }$.

Let $f\ge 0$ be Borel and choose $s_n\uparrow f$ as in [F4]. If $s_n$ is simple, its integral against $\zeta$ is a finite nonnegative linear combination of the measurable evaluations in [F1], hence is measurable. For each $\omega$, monotone convergence for the measure ${\zeta }_{\omega }$ gives ${\zeta }_{\omega }(s_n)\uparrow {\zeta }_{\omega }(f)$. The pointwise-limit result [F5] therefore makes $\omega \mapsto {\zeta }_{\omega }(f)$ measurable.

Let $(B_j{)}_{j\ge 1}$ be pairwise disjoint Borel sets. For every $\omega$, ${\zeta }_{\omega }({\bigcup }_j{B}_j)={\sum }_j{\zeta }_{\omega }(B_j)$ by [F1] and [F6]. The finite partial sums increase, so [F3] gives $\nu ({\bigcup }_j{B}_j)={\lim }_n{\sum }_{j=1}^n\mathbb{E}[\zeta (B_j)]={\sum }_j\nu (B_j)$. Together with step 1.1 this proves that $\nu$ is a measure.

For a nonnegative simple $s={\sum }_{j=1}^m{a}_j{1}_{B_j}$, the definition $\nu (B)=\mathbb{E}[\zeta (B)]$ and finite linearity give $\mathbb{E}[\zeta (s)]=\int sd\nu$. For the increasing sequence from step 1.2, both $\zeta (s_n)\uparrow \zeta (f)$ pathwise and $\int s_{n}d\nu \uparrow \int fd\nu$; applying [F3] to both nonnegative integrals proves the expectation identity.

For compact $K$, local finiteness of $\nu$ at $K$ means exactly $\nu (K)=\mathbb{E}[\zeta (K)]<\mathrm{\infty }$ by its definition. Requiring this for all compact $K$ proves the final equivalence.