"Small sets" in Markov chains
This is actually an important and much used concept in the study of Markov chains. The number $\delta$ is meaningful because one assumes $\mu$ is a probability measure (and not only a measure). Then $\delta$ is used to evaluate the rate of the loss of memory of the initial state by the chain.
If $A$ is "small," then the chain probabilistically regenerates/forgets its past once it enters $A$. Also, smallness has nothing to do with its measure or anything like that; there are many examples of chains with the entire state space that’s small.
Background: if a transition kernel is of the form $$ p^{(1)}(x,B) = \nu(B) \quad{(*)} $$ for all $x$ and $B$, then the chain elements are independent. This is because the transition doesn't depend on where it's coming from: $x$.
With a small set, we're only looking at $x \in A$. If $A$ is $k=1$ small, then $p^{(1)}(x,\cdot) \ge \delta \nu(\cdot)$ for all $x \in A$, and we can write the transition kernel as follows: \begin{align*} p^{(1)}(x,\cdot) &= \delta \nu(\cdot) + p^{(1)}(x,\cdot) -\delta \nu(\cdot)\\ &= \delta \nu(\cdot) + (1-\delta) \frac{p^{(1)}(x,\cdot) -\delta \nu(\cdot)}{1-\delta} \\ &= \delta \nu(\cdot) + (1-\delta)K(x, \cdot). \end{align*}
This is a discrete mixture, so with probability $\delta$ you're transitioning with $\nu$ and forgetting the past, and with probability $1-\delta$, you're transitioning with something that takes into account where you're coming from and not forgetting the past. The smallness property gives us the nonnegativity of $K$.
As @Did mentions, "$\delta$ is used to evaluate the rate of the loss of memory of the initial state by the chain." You can see that if $\delta = 1$, then we get equation $(*)$.
Other things: If $k > 1$, then it takes longer to forget, and if we’re talking about “petiteness” then it’s the same idea but with a random number of steps.
Yes you can find a lot related topic at Jeffery Rosenthal's paper, especially the "General State Markov chain and MCMC algorithm".
Another useful reference is Nummelin's paper called "MC for MCMC". This is more intuitively than Rosenthal and when I write paper about this topic this summer, I find it is really useful.