In the paper A Martingale approach to Infinite Systems of Interacting Processes one reads:
To understand the proof we need to operate with relations in (7.3) and (7.7).
Let' s have them here:
The definition of the resolvent is
$$R_\lambda f(\eta) = \int_0^\infty e^{-\lambda t} \Bbb{E} [f(\eta(t))]\, dt $$
we rely on the following relation stated in 7.7
for $f$ well behaved $$f(w) = \int_0^\infty e^{-w t} \frac{1}{2\pi i} \int_\Gamma e^{zt} f(z)\, dz \, dt $$
We would like to see that
$$\Bbb{E} [f(\eta(u))] = \frac{1}{2\pi i} \int_\Gamma e^{\lambda u} R_\lambda f (\eta) \, d \lambda $$
Attempt
First write
\begin{align*} \frac{1}{2\pi i} \int_\Gamma e^{\lambda u} R_\lambda f (\eta) \, d \lambda & =\frac{1}{2\pi i} \int_\Gamma e^{\lambda u} \int_0^\infty e^{-\lambda t} \Bbb{E} [f(\eta(t))]\, dt \, d \lambda \\ &= \int_0^\infty e^{-\lambda t} \frac{1}{2\pi i}\int_\Gamma e^{\lambda u} \Bbb{E} [f(\eta(t))] \, d \lambda\, dt \end{align*}
But how to conclude from here?
For the sake of completeness
Here I add the relations 7.1 (a), (c) in case one imagines this might be helpful
To prove this result, one should take the Fourrier transform of both sides of (7.9):
$$\tag{7.9} \Bbb{E}^{\Bbb{P}_\eta} [f(\eta(t))] = \frac{1}{2\pi i} \int_\Gamma e^{ z t} R_z f (\eta) \, dz $$
For the left-hand side we obtain \begin{align*} \int_0^\infty e^{-\lambda t}\Bbb{E}^{\Bbb{P}_\eta} [f(\eta(t))]\, dt = R_\lambda f(\eta) \end{align*} by the definition of $R_\lambda$. For the right-hand side we obtain
\begin{align*} \int_0^\infty e^{-\lambda t}\frac{1}{2\pi i} \int_\Gamma e^{z t} R_z f (\eta) \, dz \, dt = R_\lambda f(\eta) \end{align*} by (7.7) therefore the Laplace transform of these two functions is the same for $\lambda >1$