# understanding recurrence relation between two measure

Hi, Could anyone explain to me how he wrote $$(3.3)$$ to $$(3.4)$$, in particular, why double integration? I am convinced with $$(3.3)$$ and another thing, I do not quite understand the role of $$p(x,t)$$.

Thank you very much for helping.

The function $$p$$ is the conditional density of $$t_n$$ given $$x_n$$ (with respect to Lebesgue measure on $$[0, T]$$). That is, equation (3.2) implies that $$\Pr(t_n \in \mathcal{T} \mid x_n) = \int_{\mathcal{T}} p(x_n, u) \, du$$ for any Borel set $$\mathcal{T} \subseteq [0, T]$$.
It follows from some "usual" measure-theoretic arguments that $$\tag{1} E[g(x_n, t_n) \mid x_n] = \int_0^T g(x_n, u) p(x_n, u) \, du$$ for any sufficiently nice (e.g., bounded and measurable) function $$g : X \times [0, T] \to \mathbb{R}$$.
Now we can verify equation (3.4): \begin{aligned} E[h(x_{n+1})] &= E[h(S(x_n, t_n))] \\ &= E[E[h(S(x_n, t_n)) \mid x_n]] &&\text{double expectation theorem}\\ &= E\left[\int_0^T h(S(x_n, u)) p(x_n, u) \, du\right] &&\text{using (1) with g(x, t)=h(S(x, t))} \\&= \int_X \left[\int_0^T h(S(x, t)) p(x, t) \, dt\right] \, \mu_n(dx) &&\text{since \mu_n is the distribution of x_n} \end{aligned}