Understanding the Definition for collections of events being independent

I will put the definitions here given in Rosenthal's "A First Look at Rigorous Probability Theory" (chapter 3 section 2 in my (granted old) edition)

Anyway, Rosenthal says

a possibly-infinite collection $\{A_\alpha\}_{\alpha \in I}$ of events is said to be independent if for each $j \in N$ and each distinct finite choice $\alpha_1,\alpha_2,\dotsc, \alpha_j\in I$ we have $$\tag{3.2.1} P(A_{\alpha_1}\cap A_{\alpha_2} \cap \dots A_{\alpha_j}) = P(A_{\alpha_1})P(A_{\alpha_2}) \dots P(A_{\alpha_j})$$

Then he says

Collections of events $\{\mathcal{A_\alpha} ;\alpha \in I \}$ are independent if for all $j \in N$ , for all distinct $\alpha_1,\alpha_2,\dotsc, \alpha_j\in I$, and for all $A_1 \in \mathcal{A}_{\alpha_1},\dotsc ,A_j \in \mathcal{A}_{\alpha_j}$, equation (3.2.1) holds.

My question is whether someone can help me understand the second definition?

My current understanding is that $\mathcal{A}_\alpha$ (notice script notation) denotes collections of events, $\{ A_\alpha\}_{\alpha \in I}$, and that collections of events $\mathcal{A}_\alpha$ are independent if for any (finite?) distinct collection of events $\mathcal{A}_{\alpha_1},\dotsc, \mathcal{A}_{\alpha_j}$ (with $j\in\mathbb{N}$), If I take any one event from within each collection, those events (or I guess more specifically the collection formed by those events) satisfies $(3.2.1)$.

Is that correct?

Also, I guess what I feel most confused about is that he is talking about collections( plural) of events being independent, but he denotes it $$\{\mathcal{A_\alpha} ;\alpha \in I \}$$ which to me looks like just one collection (of whatever $\mathcal{A}_\alpha$ are). I believe that $\mathcal{A}_\alpha$ are themselves collections of events, but I don't see that stated anywhere. Is that something I am just supposed to infer or implicitly understand?

Thanks

Your confusion is justified -- the author's definitions abuse notation, at least in my opinion, leading to unnecessary ambiguity.

$I$ is the index set, this means it is the set of labels which are assigned to each event in the collection of events. Of course, each event is assigned at most one label, and each label is assigned at most one event.

The big problem here is that $I$ here is used in at least two different contexts (as is the letter $\alpha$ to denote a generic label in an index set).

Let me try to re-write the author's definitions so that they are clearer:

Independence of events within a single collection "from each other":

A possibly-infinite collection $\{A_\alpha\}_{\alpha \in I}$ of events is said to be independent if for each $i \in \mathbb{N}$ and each distinct finite choice $\alpha_1,\alpha_2,\dotsc, \alpha_i\in I$ we have $$\tag{3.2.1} P(A_{\alpha_1}\cap A_{\alpha_2} \cap \dots A_{\alpha_j}) = P(A_{\alpha_1})P(A_{\alpha_2}) \dots P(A_{\alpha_j})$$

Let's review the implicit hierarchy and try to draw an analogy -- let's say that events are like cells, collections (of events) are like human bodies. Then the notion of independence defined above is for cells within a single body.

The author's second definition is for independence of different human bodies within a society. This is why I believe it to be an abuse of notation to use the same letter $I$ to denote the index set in the first and the second definition -- in the first defintion, $I$ is indexing cells in a body, whereas in the second definition, $I$ is indexing bodies in a society. Obviously there is a clear analogy between the two, but over-using the notation leads to unnecessary blurring of the concepts.

Here I rewrite the author's second definition:

Independence of distinct collections "from each other":

Distinct collections of events $\large\{ \mathcal{A_\beta} = (\ \{A_{\alpha_{\beta}} \}_{\alpha_{\beta} \in I_{\beta}}) ;\beta \in J \}$ are independent if for all $j \in \mathbb{N}$ , for all distinct $\beta_1,\beta_2,\dotsc, \beta_j\in J$, and for all $\large A_{\alpha_{\beta_1}} \in \mathcal{A}_{\beta_1},\dotsc ,A_{\alpha_{\beta_j}} \in \mathcal{A}_{\beta_j}$, equation (3.2.1) holds.

In other words, each possible collection of events $\mathscr{C}=\{\mathscr{C}_{\beta}\}_{\beta \in J}$ formed such that $\mathscr{C}_{\beta_k} \in \mathcal{A}_{\beta_k}$ for all $\beta_k \in J$ (i.e. such that the $\beta_k$th element of $\mathscr{C}$ comes from the collection $\mathcal{A}_{\beta_k}$) is an independent collection of events in the sense of the first definition.

One can almost see why the author was imprecise because trying to be completely precise here I have almost made it somewhat less clear -- the idea is that $\beta$ denotes a generic element of $J$ which is an index, where $J$ indexes the collections, $\beta_m$ is one of $n$ arbitrary elements taken from $J$, $\alpha_{\beta}$ is a generic index of the index set $I_{\beta}$, the index set of the $\beta$th collection of events, so $I_{\beta}$ indexes events, not collections of events, $\alpha_{\beta_m}$ is an arbitrary index from the index set $I_{\beta_m}$ which is one of $n$ arbitrary index sets $I_{\beta_1}, \dots, I_{\beta_n}$, and thus $A_{\alpha_{\beta_m}}$ is an arbitrary element from the collection of events $\mathcal{A}_{\beta_m}$.

Two things to note:

1. The second definition does not imply the first one. Namely, one could have that a society consists of humans which are all independent from each other, but it is not necessarily the case that all of the cells within each human are independent from each other.

In fact, the case when the second definition holds but the first does not hold will be encountered more commonly, as is made clear in

1. The type of collections of events for which the second definition is most often applied is $\sigma-$algebras. This is obviously because $\sigma-$algebras are one of the basic components which make up a probability space; also because we need to consider multiple $\sigma-$algebras for the same event space at the same time when considering concepts like conditional expectation or stochastic processes.

For example, given $n\in\mathbb{N}$ $\sigma-$algebras $\mathscr{F}_1, \dots, \mathscr{F}_n$ it is not that atypical for all $n$ of them to be independent of each other, e.g. given $A_1 \in \mathscr{F_1}$ and $B_1 \in \mathscr{F}_2$, we have $$\mathbb{P}(A_1 \cap B_1)=\mathbb{P}(A_1)\mathbb{P}(B_1),$$ but it is usually not the case (in fact I believe it is even impossible unless the $\sigma$-algebra in question is somehow degenerate) that for all of the sets within each $\sigma-$algebra are independent from each other, i.e. usually we will have for $A_1, A_2 \in \mathscr{F}_1$ that $$\mathbb{P}(A_1 \cap A_2) \not=\mathbb{P}(A_1)\mathbb{P}(A_2).$$ This however is not a problem since the first definition does not need to hold for the second definition to hold.

1. Also, the same way how the notion of independence of distinct collections of events is used to define the notion of independent $\sigma$-algebras, the notion of independent $\sigma-$algebras is then used to define the notion of independence of random variables. Namely, two random variables $X$ and $Y$ are independent if and only if the $\sigma-$algebras generated by them, $\sigma(X)$ and $\sigma(Y)$ respectively, are independent (as $\sigma-$algebras).

Anyway, the purpose of the second definition is to define a notion of independence for $\sigma-$algebras without demanding that all of the sets within each individual $\sigma$-algebra are independent from each other (which would be the content of the first definition).