We wish to define what it means for a property to be generic for pairs of
points in Sr(α). Although Sr(α) is contained in S, the Lustig–Moriah definition of
genericity in S does not apply because the set Sr(α) is itself null in S. Nonetheless,
S1(α) = PML(Sα) is a topological sphere in its own right, and thus has its own
natural measure class. Therefore, we may define a subset E of S1(α) to be null if
E has measure zero in PML(Sα); this respects the topology of PML(Sα) sitting
inside of PML(S). We extend this notion of nullness to subsets E Sr(α) of
larger spheres in a “visual” manner by considering the set of points on the sphere
of radius 1 that are metrically between the center and E—these are the points of
S1(α) that “see” the set E.
Definition 3.6. E is null in Sr(α) if E1 := S1(α) : E Sr−1(γ) = ∅} is
null in S1(α) PML(Sα).
Remark 3.7. The definition given above is the most restrictive notion of null-
ness that makes use of S1(α) as a visual sphere (i.e., that treats the 1–sphere as
the sphere of directions). Another possible definition, also natural from the point
of view of Fubini’s theorem, would be inductive: suppose nullness has been defined
for spheres of radius 1,...,r 1. Instead of E1, the full set of points that see
E Sr(α), we form the smaller set
E1 = S1(α) : E Sr−1(γ) is not null in Sr−1(γ)}.
Then we could declare E Sr(α) to be null in Sr(α) if E1 is null in S1(α), com-
pleting the inductive definition.
Example 3.8. To get a feeling for these definitions, consider the examples of
with the Euclidean metric or
metric, with the Lebesgue measure class on the
sphere of radius 1 in each case. To accord with geometric intuition, we expect arcs
to be non-null and points to be null.
Figure 1. In each metric, an arc E and a point E are shown on
the sphere of radius two together with the associated E1 for each.
In the Euclidean metric, if E is an arc on the sphere of radius r, then E1 is also
an arc but E1 is empty. If E is a point, then E1 is a point while E1 is again empty.
In the
metric, if E is a nontrivial arc on the sphere of radius r, then E1 is a
nontrivial arc, and so is E1. In this setting, however, points in nonaxial directions
have a large E1 but an empty E1.
This means that our visual definition of nullness works intuitively in the
case (points are null but arcs are not), but less so in the
case (where even points
are typically non-null). The weaker, inductively defined, notion of nullness makes
even arcs null in Euclidean space, but on the other hand behaves intuitively on
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