Measurable Acting Group
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a measurable acting group is a special group that
acts The Acts of the Apostles (, ''Práxeis Apostólōn''; ) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message to the Roman Empire. Acts and the Gospel of Luke make up a two-par ...
on some space in a way that is compatible with structures of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
. Measurable acting groups are found in the intersection of measure theory and
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, two sub-disciplines of mathematics. Measurable acting groups are the basis for the study of invariant measures in abstract settings, most famously the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr ...
, and the study of
stationary random measure In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. D ...
s.


Definition

Let (G, \mathcal G, \circ) be a
measurable group In mathematics, a measurable group is a special type of group in the intersection between group theory and measure theory. Measurable groups are used to study measures is an abstract setting and are often closely related to topological groups. De ...
, where \mathcal G denotes the \sigma -algebra on G and \circ the
group law In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and eve ...
. Let further (S, \mathcal S) be a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
and let \mathcal A \otimes \mathcal B be the product \sigma -algebra of the \sigma -algebras \mathcal A and \mathcal B . Let G act on S with group action : \Phi \colon G \times S \to S If \Phi is a
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
from \mathcal G \otimes \mathcal S to \mathcal S , then it is called a measurable group action. In this case, the group G is said to act measurably on S .


Example: Measurable groups as measurable acting groups

One special case of measurable acting groups are measurable groups themselves. If S=G , and the group action is the group law, then a measurable group is a group G , acting measurably on G .


References

* Group theory Measure theory {{group-theory-stub