understanding $\theta,\Theta, d\theta, \mu(d\theta)$

Question

Let $(\mathcal S, d)$ be a metric space. let $\{f_\theta: \theta \in \Theta\} $ be a family of Lipschitz functions on $\mathcal S$ and $\mu$ be a probability distribution on $\Theta$. Suppose that $\int K_{\theta}\mu(d\theta)<\infty, \int d(f_{\theta}(x_0), x_0)\mu (d\theta)<\infty$ for some $x_0\in S$, and $\int \log K_{\theta}\mu(d\theta)<\infty $ Then the induced Markov chain has a unique stationary distribution. This is from here in page $47$, Theorem $1.1$ in the paper Iterated Random Functions

Now, I am not able to understand, $\theta,\Theta, d\theta, \mu(d\theta)$ here in this context. Could anyone help me to understand these four things by a little example maybe? That would so help, Thanks a lot.

For example, can I take $\Theta=\{1,2,3,4,5\}$? If yes, Then what will happen to the other three objects?

Answer 1

If you were to take $\Theta= \{1, 2, 3, 4, 5\}$ then $\theta$ would be any one of 1, 2, 3, 4, 5. That is, $\theta\in\Theta$ means that the variable $\theta$ can take on any of the values in the set $\Theta$. Since $\Theta$ is discrete, not continuous, "$d\theta$" would not be defined. If instead, we take $\Theta= (0, 1)$, the open interval from 0 to 1, then $\theta$ could be any number between 0 and 1 and $d\theta$ would be the usual differential from Calculus. I believe that "$\mu d\theta$" is the "Lebesgue measure" defined by the variable $\theta$ though I would have expected the notation "$d\mu(\theta)$.

Answer 2

To complete the previous answer, if we go to the discrete case, I suppose it is implied that: $$\int K_\theta \mu(d\theta) = \sum_{\theta \in \Theta} K_\theta \mu(\theta)$$

It is therefore not necessary to suppose that the set $\Theta$ is continuous.

Also, I believe that $\mu(d\theta)$ is simply a way to write the probability measure as a distribution that you apply to a given small element $d\theta$, which can look more intuitive I guess?

And to give a small example, say $f_\theta$ is the function on $\mathbb{R}$ such that $f_\theta (x) = \theta \times x$, then $\theta$ can indeed be any number and therefore $\Theta$ can be any set.