I am taking a graduate level measure theory class, and I have a question that is decidedly undergraduate in difficulty. I have asked myself (and others) this question in other classes and other contexts, and I have neither discovered nor been given an answer that is sufficiently satisfactory. But I am working on understanding a proof from Real Analysis for Graduate Students by Bass, and this question seems to be impossible to ignore given its relationship to the larger proof I am trying to understand. With that said, here's the question.
Given a sequence of open balls $\{B_{\alpha}\}$, what is the intended meaning of the variable $\alpha$? -Is it the cardinality of balls in the sequence? -Is it any such ball in the sequence? -Both? -Neither? Clearly, I am clueless as to how $\alpha$ is intended to be used, and the context seems to offer no hints pointing to the author's intent. Is there perhaps a generally accepted convention of understanding?
Thanks in advance to all the insight(s) provided by the respondents. Best
Formally, a sequence $\{B_\alpha\}$ is a function from an index set $\mathcal I$ to the set of all open balls $\mathcal B$, i.e. $f\colon \mathcal I\to \mathcal B$. Then $\alpha\in\mathcal I$ denotes a generic element of $\mathcal I$, and $B_\alpha$ is a way of writing $f(\alpha)$ without mentioning the function $f$. The set $\mathcal I$ is allowed to be uncountable.