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What theory are they?
Set theory comes to mind but is that too broad?
Set theory comes to mind but is that too broad?
Measure theory is a branch of mathematics that deals with the concept of measuring the size or extent of sets and their elements. It provides a rigorous framework for defining and working with concepts such as length, area, and volume in both discrete and continuous spaces.
The key concepts in measure theory include measure, measurable sets, sigma-algebras, and measurable functions. A measure is a mathematical function that assigns a number to each set, representing its size or extent. Measurable sets are sets for which a measure can be defined. A sigma-algebra is a collection of measurable sets that satisfies certain properties. Measurable functions are functions between two measurable spaces that preserve the measure of sets.
Lebesgue measure is a type of measure that is defined on all Lebesgue measurable sets, which include all Borel sets. Borel measure, on the other hand, is a type of measure that is defined only on Borel sets, which are a specific type of measurable sets. Borel sets are important because they form a sigma-algebra that is rich enough to contain most of the sets we encounter in everyday mathematics.
Measure theory has many applications in mathematics, physics, and other fields. It is used to define and study integrals, which are essential in calculus and differential equations. It also plays a crucial role in probability theory, where it is used to define the concept of probability measure. In physics, measure theory is used to define and analyze physical quantities such as energy and momentum. It also has applications in fields such as economics, computer science, and statistics.
Some important theorems in measure theory include the Carathéodory extension theorem, which states that any pre-measure on a ring can be extended to a measure on the generated sigma-algebra. The Radon-Nikodym theorem is another important result that gives necessary and sufficient conditions for the existence of a derivative of a measure with respect to another measure. Other notable theorems include the Hahn-Kolmogorov theorem and the Riesz representation theorem.