  Probabilistic databases (PDBs) are used to model uncertainty in data in a
quantitative way. In the standard formal framework, PDBs are finite probability
spaces over relational database instances. It has been argued convincingly that
this is not compatible with an open world semantics (Ceylan et al., KR 2016)
and with application scenarios that are modeled by continuous probability
distributions (Dalvi et al., CACM 2009).
  We recently introduced a model of PDBs as infinite probability spaces that
addresses these issues (Grohe and Lindner, PODS 2019). While that work was
mainly concerned with countably infinite probability spaces, our focus here is
on uncountable spaces. Such an extension is necessary to model typical
continuous probability distributions that appear in many applications. However,
an extension beyond countable probability spaces raises nontrivial foundational
issues concerned with the measurability of events and queries and ultimately
with the question whether queries have a well-defined semantics.
  It turns out that so-called finite point processes are the appropriate model
from probability theory for dealing with probabilistic databases. This model
allows us to construct suitable (uncountable) probability spaces of database
instances in a systematic way. Our main technical results are measurability
statements for relational algebra queries as well as aggregate queries and
datalog queries.
