In a variety of reasoning tasks, one estimates the likelihood of
events by means of volumes of sets they define. Such sets need to be
measurable, which is usually achieved by putting bounds, sometimes ad
hoc, on them. We address the question how unbounded or unmeasurable
sets can be measured nonetheless. Intuitively, we want to know
how likely a randomly chosen point is to be in a given set, even in the
absence of a uniform distribution over the entire space.
To address this, we follow a recently proposed approach of taking
intersection of a set with balls of increasing radius, and defining
the measure by means of the asymptotic behavior of the proportion of
such balls taken by the set. We show that this approach works for
every set definable in first-order logic with the usual arithmetic
over the reals (addition, multiplication, exponentiation, etc.), and
every uniform measure over the space, of which the usual Lebesgue
measure (area, volume, etc.) is an example. In fact we establish a
correspondence between the good asymptotic behavior and the finiteness
of the VC dimension of definable families of sets. Towards computing
the measure thus defined, we show how to avoid the asymptotics and
characterize it via a specific subset of the unit sphere. Using
definability of this set, and known techniques for sampling from the
unit sphere, we give two algorithms for estimating our measure of
unbounded unmeasurable sets, with deterministic and probabilistic
guarantees, the latter being more efficient. Finally we show that a
discrete analog of this measure exists and is similarly well-behaved.
Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces