A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.
It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure).
One important example of a measure space is a probability space.
A measurable space consists of the first two components without a specific measure.
A measure space is a triple
where[1][2] In other words, a measure space consists of a measurable space
Sticking with this convention, we set
In this simple case, the power set can be written down explicitly:
This leads to the measure space
It is a probability space, since
corresponds to the Bernoulli distribution with
which is for example used to model a fair coin flip.
Most important classes of measure spaces are defined by the properties of their associated measures.
This includes, in order of increasing generality: Another class of measure spaces are the complete measure spaces.