Moduli of path families are widely used to mark curves which may be neglected
for some applications. We introduce ordinary and approximation modulus with respect to Banach
function spaces. While these moduli lead to the same result in reflexive spaces, we show that there
are important path families (like paths tangent to a given set) which can be labeled as negligible
by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in
particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the
same space.