Stone's representation theorem states that every Boolean algebra is isomorphic to an algebra of point sets.
Loomis-Sikorski theorem states that ''every $\sigma$-complete Boolean algebra is $\sigma$-isomorphic to a $\sigma$-complete Boolean algebra of point sets modulo a $\sigma$-ideal in that algebra (Loomis, 1947)''.
Why do we have to take the quotient space of a $\sigma$-complete Boolean algebra of point sets by a $\sigma$-ideal in that algebra for the representation of $\sigma$-complete Boolean algebras?
Thanks a lot for clarifying.
