Suppose that $P(w)$ is a probability density function with support $w \in [0,\infty)$ and $G = G[P]$ is a functional satisfying $G[P] \in [0,1]$. I saw a paper used a chain rule of the following form: $$\frac{dG}{dt} = \int_0^\infty \frac{\delta G}{\delta P(w)}\,\frac{\partial P(w)}{\partial t}\,dw, $$ in which $\frac{\delta G}{\delta P(w)}$ represents the Fréchet/variational derivative of $G$ and $P(w) = P(w,t)$ is now a time-dependent probability density. May I know why such a chain rule holds? My main concern is the appearance of the integral $\int_0^\infty \,dw$.
Remark: for any sufficiently well behaved function $\eta(w)$, the relation $$\left\langle \frac{\delta G}{\delta P(w)}, \eta \right\rangle = \lim_{\epsilon \to 0} \frac{G[P+\epsilon\,\eta] - G[P]}{\epsilon} $$ defines the Fréchet derivative.