  Probabilistic databases (PDBs) model uncertainty in data in a quantitative
way. In the established formal framework, probabilistic (relational) databases
are finite probability spaces over relational database instances. This
finiteness can clash with intuitive query behavior (Ceylan et al., KR 2016),
and with application scenarios that are better modeled by continuous
probability distributions (Dalvi et al., CACM 2009).
  We formally introduced infinite PDBs in (Grohe and Lindner, PODS 2019) with a
primary focus on countably infinite spaces. 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.
  We argue that finite point processes are an appropriate model from
probability theory for dealing with general probabilistic databases. This
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.
