What is Fin(E,2)?

Question

It is easy to see that Bor(I) satisfies the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., but the proof is more difficult. (Countable chain condition, wikipedia)

So what is Fin(E,2)?

Answer 1

$\mathrm{Fin} ( E , 2 )$ is the set of all finite partial functions $E \to 2 = \{ 0 , 1 \}$, and the ordering is reverse inclusion: $p \leq q$ iff $p \supseteq q$, or, more pedantically, $\mathrm{dom} ( p ) \supseteq \mathrm{dom} ( p )$ and $p (e) = q(e)$ for all $e \in \mathrm{dom}(q)$.