The Mean-Field Cost Functional

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, let $B$ be a nonnegative real number, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $T>0$ be a real number, and let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$.

The mean-field cost of $(S,A)$ under $(L,G)$ is the real number

where the integral is the Riemann integral of $t\mapsto L(S_t,A_t)$ on $[0,T]$; this integrand is continuous on $[0,T]$, the interval regarded as a subset of the real line with the absolute value metric: componentwise continuity of $S$ and $A$ makes $t\mapsto(S_t,A_t)$ converge in Euclidean distance along every sequence $t_n\to t$ in $[0,T]$, so the sequential continuity clause of population cost data applies, and continuity follows by claims 1 and 3 of the sequential characterization of lower semicontinuity, applied to the integrand and to its negative, together with claims 1 and 2 of the negation and characterization lemma. The integral then exists by claim 3 of the integral toolkit on a compact interval.