Is there an expression for when something happens except for a particular case?

Question

In my language there is a mathematical expression for when a statement is true, except for a special subcase. It happens a lot in measure theory, for instance:

A property $P$ is said to hold "almost everywhere" in $X$ if there exists a set $N\subset X$ of measure zero such that $P$ holds for every point of $X$ except for maybe the points in $N$.

Say I wanted to rewrite that as a sentence in a simplified manner, removing the specifics of "points in" part and formulating it all around the set, like: "$P$ holds almost everywhere on $X$ if it holds on $X$ except for a set of measure zero"

However "except for" doesn't fit quite as good in this sentence as it does in my language. So I wonder: is there an expression in english that I can use it to rewrite this simplified version? Please, keep in mind the measure thing is just an example, I'm not looking for a way of stating this specific expression, I want a mathematical expression to shorten the sentences like I said in this example, but that should hold for similar cases as well.

Answer 1

Sometimes you can use the prefix co- to describe the bigness of what remains when the exceptional set is neglected. Thus, there is co-finite (exceptional set is finite), co-countable (exceptional set is countable), co-meager (exceptional set is meager/first-category), co-null (exceptional set is "null" in some sense, often used for measure zero, but I tend to avoid it because "null" often means empty set), etc.

Another common method is to use the term typical element or generic element to describe an arbitrarily chosen non-exceptional element. However, the terminology doesn’t allow us to specify what type of smallness the exceptional set satisfies (but see the next paragraph for a way to fix this), and in practice it’s primarily used when the exceptional set is first category. See the first few paragraphs of my answer to Generic Elements of a Set for the likely origin of the use of typical in this context.

You can also use almost every or typical along with an appropriate modifier word/term to describe the bigness (I invented this many years ago as a useful shorthand -- for example, see here -- although it's very likely others have independently invented this wording), such as Lebesgue almost every or Lebesgue typical (exceptional set is Lebesgue measure zero), Baire almost every or Baire typical (exceptional set is first category), porosity typical (exceptional set is $\sigma$-porous, or maybe even porous depending on context), etc.

Consistent with the above, I’ve seen the expression $\mathcal{I}$-typical used when the exceptional set belongs to some $\sigma$-ideal $\mathcal{I}$ of sets.

In the case where the exceptional set has measure zero, there is a fairly common term, namely full measure, or full $\mu$-measure when you want to specify the measure $\mu.$