I was reading here about the general definition of independent events. The definition stated there is
Independent Events: Let $\mathcal{E}$ be an experiment with probability space $(\Omega, \Sigma, Pr)$. Let $\mathcal{A} = \{A_i : i\in I\}$ be a family of events of $\mathcal{E}$. Then $\mathcal{A}$ is independent if and only if $\,\,\forall \,\,\text{finite} \,\,J\subseteq I$ $$Pr\left(\bigcap_{i\in J}A_i\right)=\prod_{i\in J}Pr(A_i)$$
The part that I am concerned about is the initial statement, not the consequence, which I am fine with. The text talks about a family of events being independent. However, always on proofwiki here is the definition of a family (see here):
Indexing Set: Let $I$ and $S$ be sets. Let $x: I\longrightarrow S$ be a function. The set $I$ of $x$ is called indexing set of the sequence $\langle x_i \rangle_{i\in I}$.
Family: The codomain $S$, consisting of the terms $\langle x_i\rangle_{i\in I}$, along with the indexing function $x$ itself, is called a family of elements of $S$ indexed by $I$.
So actually, a family is not just a sequence! But it is a sequence PLUS the indexing function. Then is the "independent events" definition saying that "sequence + indexing function" are independent? Or is it a mistake and they actually meant that only the sequence is independent? Here are my suggestions:
Independent Events (only sequence): Let $\mathcal{E}$ be an experiment with probability space $(\Omega, \Sigma, Pr)$. Let $\mathcal{A} = \langle A_i \rangle_{i\in I}$ be a sequence of events of $\mathcal{E}$ indexed by the function $x: I\subseteq \mathbb{N} \longrightarrow \Sigma$. Then the sequence $\mathcal{A}$ is independent if and only if $\,\,\forall \,\,\text{finite} \,\,J\subseteq I$ $$Pr\left(\bigcap_{i\in J}A_i\right)=\prod_{i\in J}Pr(A_i)$$
Independent Events (family): Let $\mathcal{E}$ be an experiment with probability space $(\Omega, \Sigma, Pr)$. Let $\mathcal{A} = \langle A_i \rangle_{i\in I}$ be a sequence of events of $\mathcal{E}$ indexed by the function $x: I\subseteq \mathbb{N} \longrightarrow \Sigma$. Then the family $(\mathcal{A}, x)$ is independent if and only if $\,\,\forall \,\,\text{finite} \,\,J\subseteq I$ $$Pr\left(\bigcap_{i\in J}A_i\right)=\prod_{i\in J}Pr(A_i)$$