$\sigma(x_t)$ is increasing for $(x_t)_{t\in \mathbb N}$ a discrete time Markov chain on $\mathbb R$

Question

I would like to show that $\sigma(x_{t-1})\subseteq\sigma(x_t)$ where $(x_t)_{t\in \mathbb N}$ is a discrete time Markov chain with state-space $\mathbb R$ and $\sigma(x_t)$ denotes the smallest $\sigma$-algebra such that $x_t$ is measurable.

Any help would be appreciated.

Answer 1

This is false. Any independent sequence $(x_t)$ is a Markov Chain and in this case $\sigma(x_{t-1}) \subset \sigma (x_t)$ is not true unless $x_{t-1}$ is a constant.