Measure Theory
Introduction to measure theoryMeasure theory is the study of measures, aiming to formalize the notion of size.
A measure is defined as a function that assigns a non-negative real number to subsets of a set . Measures must be countably additive, meaning the measure of a set that can be decomposed into a finite or countably infinite number of ‘smaller’ subsets must be equal to the sum of the measures on each of the smaller subsets. Formally, a measure is a function mapping from a -algebra over a set to the extended real number line satisfying the following three properties:
- Non-negativity:
- Null empty set:
- Countable additivity:
Note the axiom of countable additivity is also an axiom of probability theory. In order to assign a consistent measure to all possible subsets of a set, it is typically the case that only trivial measures (like the counting measure) satisfy the necessary properties. This issue is resolved by only defining a measure on a subcollection of all possible subsets; the subsets called measurable subsets, which necessarily form a -algebra.
Measurable Space
The pair is a called a measurable space, and the elements of are known as measurable sets. Note that, unlike a measure space, no measure need be defined (in accordance with its name); it is simply measurable (thanks to ) meaning a measure can be defined on the space, but refrains from defining one explicitly.
Measure Space
A measure space is a triple where
- is a non-empty set
- is a -algebra on
- is a measure defined on the measurable space
A measure space includes the necessary components (a measure and a measurable space) for consistently defining a measure in an appropriate context.
-algebra
Formally, a -algebra on a set is a collection of subsets of (including itself) that are closed under complement and countable unions. That is, the complement of any subset in is also in , and the union of any countable number of subsets in is also in . A -algebra is a type of algebra of sets, and is useful in measure theory as the collection of subsets for which a measure is defined (measureable sets) is necessarily a -algebra.
Example
For the set , an example of a -algebra on is
Notice that any element of has its complement in , and any countable union of the elements of is also in .