I am curious why a in measure theory often the notation $A_1,A_2,\ldots$ is preferred to $(A_n)_n$.
For example we can say a measure space $(A,B,\mu)$ is sigma finite if there exist a sequence $(A_n)_{n\in\mathbb N}$ of measurable sets with finite measure with $\bigcup_{n=1}^\infty A_n=A$.
On the other hand one can say $(A,B,\mu)$ is sigma finite if there are at most countably many sets $A_1,A_2,\ldots \in B$ with $\mu(A_n)<\infty $ for all $n\in\mathbb N$ that satisfy $\bigcup_{n=1}^\infty A_n=A$.
I guess both definitions ar equivalent, but the latter is often preferred. Why?