a probability measure $\mu$, finiteness of the integral above is equivalent to $\mu \in {\mathcal P}_1(E)$, whenever $E$ is endowed with the distance induced by the norm $\lVert\cdot  \rVert$.

We now recall the definition of push-forward measure.

\[pushforward\] Let $\mu\in\mathcal{M}_+(X)$ and $f  \colon X\to Z$ a $\mu$-measurable function be given. The push-forward measure $f_\#\mu\in\mathcal{M}_+(Z)$ is defined by $f_\#\mu(B)=\mu(f^{-1}(B))$ for any Borel set $B\subset Z$. The push-forward measures has the same total mass as $\mu$, namely $\mu(X)=f_\#\mu(Z)$. It also holds the change of variables formula $$\int_Z g\,\mathrm{d}f_\#\mu=\int_X g\circ f\,\mathrm{d}\mu$$ whenever either one of the integrals makes sense.

For $E$ being a Banach space, the notation $C^1_b(E)$ will be used to denote the subspace of $C_b(E)$ of functions having bounded continuous Fréchet differential at each point. The notation $D\phi(\cdot)$ will be used to denote the Fréchet differential. In the case of a function $\phi \colon [0,T]\times E \to \mathbb{R}$, the simbol $\partial_t$ will be used to denote partial differentiation with respect to $t$, while $D$ will only stand for differentiation with respect to the variables in $E$.

Functional setting
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The space of labels $U$ will be assumed to be a compact metric space. The state of the system is described by 