Reasoning about Measures of Unmeasurable Sets

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.