Notation
For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a history $h^{t}$ occurs and $\Pi(h^{t}|h^{t-1})$ denote the probability that a history $h^t$ follows from a history $h^{t-1}$. Moreover, let $a_t$ and $b_t$ be functions that take an element of $H^t$ into $\mathbb{R}$.
Question
Assuming that both series converge, does the following equality hold?
$$\sum_{t=1}^{\infty}\sum_{h^{t-1}}\Pi(h^{t-1})a_{t-1}(h^{t-1})\sum_{h^{t}|h^{t-1}}\Pi(h^{t}|h^{t-1})b_t(h^{t})=\sum_{t=1}^{\infty}\sum_{h^{t}}\Pi(h^{t})a_{t-1}(h^{t-1})b_t(h^{t})$$
Comments
On one hand, it seems to me that it follows from the fact that
$$\sum_{h^{t-1}}\Pi(h^{t-1})\sum_{h^{t}|h^{t-1}}\Pi(h^{t}|h^{t-1})=\sum_{h^{t}}\Pi(h^{t})$$
since $\Pi(h^{t-1})\Pi(h^{t}|h^{t-1})=\Pi(h^{t})$. On the other hand, the right hand side has a summation over $h^t$ but $h^{t-1}$ shows up in it which appears to be incorrect to me.