"non-zero" is not the same as a "all but a set of measure zero". Here's what "measure zero" means:

A set S of real numbers has measure zero if for any positive ε no matter how small, there exists a countable set of intervals such that (1) every element of S is in at least one of the intervals, and (2) the total length of the intervals is < ε.

For example, let S be the set of positive integers, {1, 2, 3, ...}. Proof: consider the set of intervals {I_1, I_2, I_3, ...}, where I_n is the interval [n-ε/2^(n+2), n+ε/2^(n+2)]. Every member of S is contained in one of these intervals.

The length of I_n is ε/2^(n+1). The length of all the intervals is ε(1/4 + 1/8 + 1/16 + ...) = ε/2 which is < ε.

Thus S, the set of positive integers, has measure zero.

A similar argument works for any countable set of real numbers, such as the rational numbers or the algebraic numbers, and so something that was true everywhere except at rational numbers would by true for "almost all" real numbers.