IN Rogers & Williams "Diffusions, Markov Process and Martingales" they introduce the resolvent as:
$$R_\lambda f(x):=\int_{[0,\infty)}e^{-\lambda t}P_tf(x)dt=\int_ER_\lambda(x,dy)f(y)$$
where $P_t$ is a measurable transition function, $R_\lambda(x,\Gamma):=\int_{[0,\infty)}e^{-\lambda t}P_t(x,\Gamma)dt,\Gamma\in \mathcal{E}$ and $f\in b\mathcal{E}$ of a measurable space $(E,\mathcal{E})$. In addition we can suppose that the map $(t,x)\mapsto P_t(x,\Gamma)$ is $\mathcal{E}\times \mathcal{B}([0,\infty))$ measurable. Right after these equations they write: "Trivial application of monotone-class theorems and of Fubini's theorem are now made without comment."
For the equation above you have to use Fubini. Furthermore, I think for the claim $R_\lambda:b\mathcal{E}\to b\mathcal{E}$ you have to use a monotone class argument. Using measuretheoretic induction, we can prove that $P_t:b\mathcal{E}\to b\mathcal{E}$. So $P_tf$ is again bounded and measurable. To conclude we must show that
$$\int_{[0,\infty)}e^{-\lambda t}P_tf(\cdot)dt\in b\mathcal{E}$$
given $P_tf(\cdot)\in b\mathcal{E}$. However here my problem starts. I need to find a class $\mathcal{H}$ of bounded functions on a set $S$ into $\mathbb{R}$ such that:
- $\mathcal{H}$ is a vector space over $\mathbb{R}$
- the constant function $1$ is an element of $\mathcal{H}$
- if $\{f_n\}$ is a sequence of non-negative functions in $\mathcal{H}$ such that $f_n\uparrow f$, where $f$ is a bounded function on $S$, then $f\in \mathcal{H}$.
Suppose futher that $\mathcal{H}$ contains the indicator of every set in some $\pi-$system $\mathcal{C}$. Then $\mathcal{H}$ contains every bounded $\sigma(\mathcal{C})-$measurable function on $S$.
So the question is: Is this the point where we have to use a monotone class argument to ensure that $R_\lambda$ is a map on $b\mathcal{E}$? And if so, how to choose $\mathcal{H}$?