Good way to state, in english, the negation of "$f$ equals zero almost everywhere".

Question

Let $X = (X, \mathcal{E}, \mu)$ be a measure space and let $f$ be a measurable function on $X$. Consider the statement:

$f$ equals zero almost everywhere

Is there an concise, unambiguous way of stating the negation of this statement? One shouldn't really just say something like

$f$ does not equal zero almost everywhere

because this could be interpreted as meaning that the complement of the set $\{x : f(x) \neq 0\}$ has measure zero. The best I've been able to come up with are variations on

It is not the case that $f$ equals zero almost everywhere

but this seems a bit clunky to me. Any advice?

Answer 1

Some people like to talk about a function's "support". In these terms the property you state can be concisely stated as

$f$ has support of positive measure

or maybe

$f$ has nontrivial support

You could also just interchange the words a bit to get

$f$ is not almost-everywhere zero

which is unambiguous (but does sound a little funny).

Answer 2

My vote is

$f$ does not vanish almost everywhere.

I suppose that could be misinterpreted in the same way as "$f$ is not zero almost everywhere" but somehow that seems much less likely with this version; I can't define precisely why.

Answer 3

One concise way of expressing this is to say

$f$ is nonzero on a set of positive measure

Alternatively, you could try

The zeros of $f$ do not have full measure

though I prefer the first.