In my lecture notes for the probability course I am taking the word "arbitrary" appears a lot to refer to sequences of events. But I'm not sure what exactly this means. In a supplementary section, I have a lemma which reads:
"Let $(A_a)_{a \in \mathcal{A}}$ and $(B_b)_{b \in \mathcal{B}}$ be arbitrary, countable or uncountable, collections of sets..."
Which leads me to think that arbitrary means countable or uncountable. But in the first chapter I have an example saying:
"If $(A_n)_{n \geq 1}$ is a sequence of arbitrary events in some probability space $(\Omega, \mathcal{F}, P)$, then the sequence $(B_n)_{n \geq 1}$ with $B_n = \bigcup_{k=1}^n A_k$ is increasing."
The index of the union at the end suggest to me that the $A_k$ must be countable in this case, otherwise I don't see how this makes sense. So what exactly does arbitrary mean?