This paper I was reading to understand how to find an invariant measure of discrete-time Markov chain with uncountable state-space. The article deals with Markovian shape changes in rectangles and finds invariant distribution. I read several times and understand that they have divided three different ways to select the next rectangle in the sequence. Their strategy is not fully clear to me too. Why do they assume that the shape changes in rectangles are Markovian? How to prove that?
Could anyone have time to look and let me know if $\bar{\mathscr L}$ means the complement of the event $ {\mathscr L}$? And how to understand the following part? Thank you.
\begin{aligned} \int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau=& \int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau, \mathscr{E}\} \mathbb{P}\{\mathscr{E} \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau \\ &+\int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau, \overline{\mathscr{}}\} \mathbb{P}\{\overline{\mathscr{}} \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau \\ =& \int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau, \mathscr{E}\} \mathbb{P}\{\mathscr{E} \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau \\ &+\int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, 1-\tau, \mathscr{E}\} \mathbb{P}\{\mathcal{E} \mid y, \mathscr{L}, 1-\tau\} \mathrm{d} \tau \\ =& \int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau, \mathscr{E}\} \mathbb{P}\{\mathscr{E} \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau \\ &+\int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \eta, \mathscr{E}\} \mathbb{P}\{\mathscr{E} \mid y, \mathscr{L}, \eta\} \mathrm{d} \eta \\ =& 2 \int_{0}^{1} \mathbb{P}\{X \leq x \mid y, \mathscr{L}, \tau, \mathscr{E}\} \mathbb{P}\{\mathcal{E} \mid y, \mathscr{L}, \tau\} \mathrm{d} \tau . \end{aligned}