Let $(\Omega,\mathcal{A},\mu)$ a measure space. Let $I$ be some $\sigma$-ideal in $(\Omega,\mathcal{A})$, i.e. it contains the empty set and contains subsets and countable unions of its elements.
How do you define $\Omega/ I$ ? And what are the equivalence classes defined as?
I am just familiar with quotient space of vector spaces, but not with that general case. Any literature is welcome aswell.
In the standard group theoretic context, $G/H$ is the equivalence classes of the relation $a\cdot b^{-1}\in H$ (consider them as right cosets here). If the group is abelian, it simply means that $a-b\in H$, and the cosets are exactly $a+H=\{a+b\mid b\in H\}$.
In the ring theoretic case, this is also how we define the equivalence relation defined by the ideal.
Note that a $\sigma$-algebra, indeed any algebra of sets, is a ring of characteristics $2$ with $\triangle$, symmetric difference, as the addition and $\cap$ as multiplication.
So naturally, $A\sim_I B$ if and only if $A\mathbin{\triangle} B\in I$. And the equivalence class of $A$ is $[A]_I = \{A\mathbin{\triangle} X\mid X\in I\}$.