Supermeasure is a term used in mathematics to describe a specific kind of measure that generalizes the concept of a measure on a measurable space. It is primarily used in the field of set theory and related areas of mathematics.
Formally, a supermeasure is defined as a set function on a sigma-algebra, which assigns a non-negative real number to each set, including those that may not be measurable. Unlike a regular measure, a supermeasure can assign "infinite" or "undefined" values to sets that do not have a well-defined measure.
Supermeasures have several important properties. Firstly, they are countably additive, meaning that the supermeasure of a countable union of pairwise disjoint sets is equal to the sum of their individual supermeasures. However, it is not required to be defined on the entire sigma-algebra, as it can assign infinite or undefined values to certain sets.
Supermeasures are often used to study non-standard or generalized measures in mathematics, allowing for a more flexible and expansive understanding of measure theory. They have applications in areas such as probability theory, set theory, and functional analysis.
In summary, a supermeasure is an extension of the concept of a measure that allows for assigning non-standard values to sets, including infinite or undefined values. It is a fundamental tool in the study of general measures and their applications in various branches of mathematics.
