Equivalent of "almost everywhere" for "holds except on a set that is nowhere dense"

Question

When we say that a condition holds everywhere except on a set of measure zero, we can say that the condition holds almost everywhere.

I want to say that a condition holds everywhere except on a set that is nowhere dense (i.e. a set that does not contain an interval). What is the "almost everywhere" equivalent shorthand notation for saying this? "Approximately almost everywhere" is my own name for it, but I am wondering if a formal name exists.

Answer 1

It is common to say that a property holds "generically" if it holds except on a meager set. A meager set is a countable union of nowhere dense sets - because meager sets and sets of measure zero are both closed under countable unions, meager is a better analogue of measure zero than nowhere dense (for most purposes).

Answer 2

No, almost everywhere is not everywhere except on a set that is nowhere dense. The reason is that nowhere dense sets could have positive Lebesgue measure, e.g. fat Cantor set. In general, there is no condition that holds everywhere except on a nowhere dense set because a nowhere dense set can have any positive measure.