I'm working through Billingsley's Probability and Measure, 2nd edition for a class, and I realized that I'm unclear about the definition that he gives for $\mathscr{A}$-equivalence on page 57 and page 58. He states:
The notion of partial information can be looked at in terms of partitions. Say that points $\omega$ and $\omega'$ are $\mathscr{A}$-equivalent if, for every $A$ in $\mathscr{A}$, $\omega$ and $\omega'$ lie either both in $A$ or both in $A^c$ -- that is, if $$I_A(\omega) = I_A(\omega'), A \in \mathscr{A}$$ This relation partitions $\Omega$ into sets of equivalent points; call this the $\mathscr{A}$-partition.
However, I'm actually a little confused about what elements would be members of an equivalence class. If I had a set {$\emptyset$, {1,2}, {3,4}, {1,2,3,4}}, we have 1 $\sim$ 2 and 3 $\sim$ 4, and the negation in other cases. So would this form 2 equivalence classes {1,2} and {3,4}? Or, since it is the case that for 1, 2 and 3, 4 both number are either in $A$ or both in $A^c$ would we say that this forms a single equivalence class {1,2,3,4}? Or something else entirely?
Nevermind, I'm just not that familiar with equivalence classes and relations and was just getting confused on some basic definitions. We would have to select and $\omega$ for our equivalence relation to be on, so, in fact, we could form 2 equivalence classes from the set I posted. One equivalence class would be formed on the number 1 (or 2) and the other would be formed on the number 3 (or 4).