Let $N=(N_t)_{t\geq 0}$ be a Poisson process with jumping times $(T_i)_{i\in \mathbb N}$ and intensity $\lambda\in L^1_{loc}([0,\infty))$, i.e. \begin{align} E[N_t]=\int_0^t \lambda(s)ds, \forall t\geq 0. \end{align} The Stieltjes integral with respect to $N$ is defined by, for $f: [0,\infty)\longrightarrow \mathbb R$, \begin{align} \int_0^\infty f(t)dN_t:=\sum_{i=1}^\infty f(T_i) 1_{T_i\leq +\infty}. \end{align} How can I prove that \begin{align*} E[\int_0^\infty f(t)dN_t] = \int_0^\infty f(t)\lambda(t) dt. \end{align*}
I have no ideas to prove it because I don't know how to link it with the first identity. Any idea is appreciated.