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, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and positive operator-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. According to Thomas W. Hawkins Jr., "It was primarily through the theory of multiple integrals and, in particular the work of Camille Jordan that the importance of the notion of measurability was first recognized."
Definition
Let X {\displaystyle X} be a set and Σ {\displaystyle \Sigma } a σ-algebra over X {\displaystyle X} , defining subsets of X {\displaystyle X} that are "measurable". A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the interval [ 0 , + ∞ ] {\displaystyle [0,+\infty ]} , that is, the non-negative real number line together with new (so-called infinite) value + ∞ {\displaystyle +\infty } , used to denote elements greater than all other (so-called finite) elements, is called a measure if the following conditions hold:
μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0}
Countable additivity (or σ-additivity): For all countable collections { E k } k = 1 ∞ {\displaystyle \{E_{k}\}_{k=1}^{\infty }} of pairwise disjoint sets in Σ {\displaystyle \Sigma } , μ ( ⋃ k = 1 ∞ E k ) = ∑ k = 1 ∞ μ ( E k ) {\displaystyle \mu {\left(\bigcup _{k=1}^{\infty }E_{k}\right)}=\sum _{k=1}^{\infty }\mu (E_{k})}
If at least one set E {\displaystyle E} has finite measure, then the requirement μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} is met automatically due to countable additivity: μ ( E ) = μ ( E ∪ ∅ ) = μ ( E ) + μ ( ∅ ) , {\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),} and therefore μ ( ∅ ) = 0. {\displaystyle \mu (\varnothing )=0.}
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