I am reading the book Discrete-Time Markov Control Processes: Basic Optimality Criteria by Onésimo Hernández-Lerma and Jean Bernard Lasserre. On page 14 of the book, they define a Markov control model which is the object studied throughout the book. The state space is denoted by $X$ and and is assumed to be a Borel space throughout the book. They define a Borel space in several different spots (e.g., page xiv, page 13, page 169, etc). I realized that I did not fully understand their definition of a Borel space so I figured I would ask here. Here are the first two paragraphs on page 169 of the book:
A topological space will always be endowed with the Borel $\sigma$-algebra $\mathcal{B}(X)$, that is, the smallest $\sigma$-algebra of subsets of $X$ that contains all of the open sets in $X$. Thus, when referring to either sets of functions, "measurable" means "Borel-measurable."
A Borel subset of a complete and separable metric space is called a Borel space. A Borel subset of a Borel space is itself a Borel space. Examples of Borel spaces are…"
How it is read above, the definition translates to this: "Let $(Y,d)$ be a complete and separable metric space. We call any $X\in\mathcal{B}(Y)$ a Borel space."
There are two issues I have with this definition. First, the use of the word "space" typically means there is some additional structure on it, i.e., a $\sigma$-algebra, topology, metric, etc. But, as stated, there is no requirement of any additional structure. My assumption is that some additional structure (most likely an associated $\sigma$-algebra) is implicit in their definition. Second, they do not go on to explain why $X$ being subset of both a "complete" and "separable" metric space is important.
My guess is that the definition should really be the following: "Let $(Y,d)$ be a complete and separable metric space, $X\in\mathcal{B}(Y)$, and $\mathcal{B}(X):=X\cap \mathcal{B}(Y)$. We call the measurable space $(X, \mathcal{B}(X))$ a Borel space."
This new definition resolves my first concern, but I still do not have good insight into why "complete and separable" is important. After all, a subspace of a complete metric spaces must be closed in order to be complete as well. So there is no guarantee that $X$ is complete in its own right. (Subspaces of separable metric spaces are separable, however.)
Wikipedia says some things about "standard Borel spaces," see this link. But I couldn't really connect this to my question. I also did a literature search on Google Scholar. I found some articles that use the same definition as given in this book but without additional explanation. Hoping to get a more proper definition and explanation so that I can proceed with my reading.
Best Answer
You said yourself that a topological space is by default endowed with its Borel $\sigma$-algebra, so yes when one says that $X$ is a Borel space it should be understood $(X,\mathcal B(X))$ is a Borel space.
I did not read the book, but since it speaks of Markov processes I assume it uses conditional distribution. So for random variables $X$ and $Y$ valued in a space $E$, they would need to be able to consider the conditional distribution of $Y$ given $X$. But the existence of such a thing requires some assumption on $E$. For example the existence is guaranteed when $E$ is a Borel subset of a Polish space.
For the record, the most general assumption on $E$ I know for the conditional distribution to exist would be that there exists a measurable isomorphism (that is a measurable bijection whose inverse is also measurable) between $E$ and a Borel subset of $\mathbb R$. It happens that any Borel subset of a Polish space satisfies this, so we can restrict ourselves to this kind of spaces. In practice it is enough for most applications.