and Convergence in Category by Arnold

Suppose that S ⊆ P (X) is a σ-field of subsets ofX and I ⊆ S is a σ-ideal. If I has the countable chain condition (ccc), i.e., every family of disjoint sets in S \ I is countable, then S/I is a complete boolean algebra. A boolean algebra is atomic iff there is an atom beneath every nonzero element. A function f : X → R is S-measurable iff f(U) ∈ S for every open set U . A sequence of S-measurable functions fn : X → R converges I-a.e. to a function f iff there exists A ∈ I such that fn(x) → f(x) for all x ∈ (X \ A). If (X,S, μ) is a finite measure space, then a sequence of measurable functions fn : X → R converges in measure to a function f iff for any ǫ > 0 there exists N such that for any n > N :