Expectation of a Markov chain

Question

Suppose I have a Markov chain $\{X_n\}_{n\in\mathbb N}$ with state space in the interval $[0,1]$. Suppose that for all $A\subseteq [0,1]$ it holds that \begin{align} P(X_n\in A|X_{n-1})=\int_AJ(X_{n-1}, y)dy, \end{align} where $J:[0,1]\times [0,1]\to \mathbb R$ is continuous symmetric and such that $\int_0^1J(x,y)dy=1$ for all $x\in [0,1]$. Let $h:[0,1]\to \mathbb R$. Can I write the following equality? \begin{align} \mathbb E(h(X_n)\mathbb I_{(X_n\in A)})=&\int_0^1h(X_n)\mathbb I_{(X_n\in A)}P(dX_n)\\ =&\int_0^1\Big(\int_Ah(y)J(X_{n-1}, y)dy\Big)P(dX_{n-1}) \end{align}