I would use the setting of discrete time and finite state space, where it has less technicalities and I am more comfortable myself.

Starting with an example, I would suggest a family of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ with values in $\{0,1\}$ with the property that the conditional probability of $X_t$ given the past depends on the weighted density of $1$s seen in the past, where $k$ step into the past is weighted with $2^{-k}$. More specifically,

$$\mathbb{P}(X_t=1\,|\,X_{t-1},X_{t-2},\ldots)= \frac{1}{2}X_{t-1}+\frac{1}{4}X_{t-2}+\frac{1}{8}X_{t-3}+\cdots\;.$$

This is not a Markov process (of any order), because of the dependence on the states arbitrarily far in the past. But this dependence becomes more and more negligible as we go farther to the past, and this we call the Feller property.

I would then go on with an abstract process with finite state space $S$ and introduce a transition kernel $K(\cdot,\cdot)$ that given values for $(\cdots,X_{t-2},X_{t-1})$, returns a distribution for $(\cdots,X_{t-1},X_t)$. More specifically, let $\mathbf{X}_t:=(\cdots,X_{t-1},X_t)$ be the history of the process up to time $t$ and denote by $\mathcal{H}=\{(\cdots,x_{-1},x_0): x_i\in S\}\cong S^{-\mathbb{N}}$ the space of all possible histories. Then,

$$\mathbb{P}(\mathbf{X}_t\in E\,|\,\mathbf{X}_{t-1})=K(\mathbf{X}_{t-1},E)$$

almost surely, for every measurable set $E\subseteq\mathcal{H}$. Here, I consider $\mathcal{H}$ as a topological space with the product topology and use the Borel $\sigma$-algebra on it. The transition kernel $K$ is Feller if it is continuous on its first argument.

Finally, I would point out how a transition kernel acts on probability measures on $\mathcal{H}$ (from the right) and on bounded measurable functions $f:\mathcal{H}\to\mathbb{R}$ (from the left), in analogy with the transition matrix of a Markov chain. I would demonstrate (or ask the students to argue) that the Feller property is equivalent to either of the following properties:

• $\pi\mapsto\pi K$ is continuous on probability measures on $\mathcal{H}$ with weak topology;

• $f\mapsto K f$ maps every continuous function $f:\mathcal{H}\to\mathbb{R}$ to a continuous function.

I would use the setting of discrete time and finite state space, where it has less technicalities and I am more comfortable myself.

Starting with an example, I would suggest a family of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ with values in $\{0,1\}$ with the property that the conditional probability of $X_t$ given the past depends on the weighted density of $1$s seen in the past, where $k$ step into the past is weighted with $2^{-k}$. More specifically,

$$\mathbb{P}(X_t=1\,|\,X_{t-1},X_{t-2},\ldots)= \frac{1}{2}X_{t-1}+\frac{1}{4}X_{t-2}+\frac{1}{8}X_{t-3}+\cdots\;.$$

This is not a Markov process (of any order), because of the dependence on the states arbitrarily far in the past. But this dependence becomes more and more negligible as we go farther to the past, and this we call the Feller property.

I would then go on with an abstract process with finite state space $S$ and introduce a transition kernel $K(\cdot,\cdot)$ that given values for $(\cdots,X_{t-2},X_{t-1})$, returns a distribution for $(\cdots,X_{t-1},X_t)$. More specifically, let $\mathbf{X}_t:=(\cdots,X_{t-1},X_t)$ be the history of the process up to time $t$ and denote by $\mathcal{H}=\{(\cdots,x_{-1},x_0): x_i\in S\}\cong S^{-\mathbb{N}}$ the space of all possible histories. Then,

$$\mathbb{P}(\mathbf{X}_t\in E\,|\,\mathbf{X}_{t-1})=K(\mathbf{X}_{t-1},E)$$

almost surely, for every measurable set $E\subseteq\mathcal{H}$. Here, I consider $\mathcal{H}$ as a topological space with the product topology and use the Borel $\sigma$-algebra on it. The transition kernel $K$ is Feller if $K(\cdot,E)$ is continuous for every cylinder set $E=\{(\cdots,x_{-1},x_0)\in\mathcal{H}: x_i=w_i \text{ for $i=0,-1,\ldots,-n$}\}$.

Finally, I would point out how a transition kernel acts on probability measures on $\mathcal{H}$ (from the right) and on bounded measurable functions $f:\mathcal{H}\to\mathbb{R}$ (from the left), in analogy with the transition matrix of a Markov chain. I would demonstrate (or ask the students to argue) that the Feller property is equivalent to either of the following properties:

• $\pi\mapsto\pi K$ is continuous on probability measures on $\mathcal{H}$ with weak topology;

• $f\mapsto K f$ maps every continuous function $f:\mathcal{H}\to\mathbb{R}$ to a continuous function.