Let $(X,ℳ,μ)$ be a measure space and $μ^*$ the external measure induced by $μ$. Which one of the following definition of "almost everywhere" (a.e.) is preferable, from both the points of view of logical correctness and usefulness?
1) Let $P(⋅)$ be a property of subsets of $X$. We say "$P(A) \ a.e.$" if $A ⊆ X$ and there is $E ⊆ X$ such that $P(A-E)$ and $μ^* (E) = 0$.
2) Let $P(⋅)$ be a property of measurable subsets of $X$. We say "$P(A)\ a.e.$" if $A ⊆ X$ and there is $A_0 ∈ ℳ$ such that $P(A_0)$, $A_0 \subseteq A$ and $μ^*(A-A_0) = 0$.
3) Let $P(⋅)$ be a property of subsets of $X$. We say "$P(A) \ a.e.$" if $A ⊆ X$ and there is $E ∈ ℳ$ such that $P(A-E)$ and $μ(E) = 0$.
4) Let $P(⋅)$ be a property of measurable subsets of $X$. We say "$P(A) \ a.e.$" if $A ∈ ℳ$ and there is $E ∈ ℳ$ such that $P(A-E)$ and $μ (E) = 0$.
I was propending for def2 or def4, because $P(A)$ is usually something like "$f(x)$ is such and such for every $x ∈ A$", and the purpose is to derive some characteristics of $∫_Af$, which doesn't make sense if $A ∉ ℳ$. But 1) is more general and I don't see any immediate contraindication, while 2) focuses on the exception set like most authors do.
In the context of Filters (see https://en.wikipedia.org/wiki/Filter_(mathematics)) there is a similar notion that may provide insight. In short, a filter is a collection of subsets one considers "large." E.g. the collection of all subsets with full measure.
Let $X$ be a set. Let $\mathcal{F}$ be a filter in $X$. Let $\varphi$ be a property with one free variable $x$.
Say $\mathcal{F}$-almost-every $x\in X$ satisfies $\varphi$ if $\{x\in X:\varphi(x)\}\in\mathcal{F}$.