6 SPENCER DOWDALL, MOON DUCHIN, AND HOWARD MASUR
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
R2
with the Euclidean metric or
1
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
1
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
2
case (points are null but arcs are not), but less so in the
1
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
1.
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