Measure (mathematics)

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 projection-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."

Let ${\displaystyle X}$ be a set and ${\displaystyle \Sigma }$ a σ-algebra over ${\displaystyle X}$, defining subsets of ${\displaystyle X}$ that are "measurable". A set function ${\displaystyle \mu }$ from ${\displaystyle \Sigma }$ to the extended real number line, that is, the real number line together with new (so-called infinite) values ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$, respectively greater and lower than all other (so-called finite) elements, is called a measure if the following conditions hold:

If at least one set ${\displaystyle E}$ has finite measure, then the requirement ${\displaystyle \mu (\varnothing )=0}$ is met automatically due to countable additivity:$${\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),}$$and therefore ${\displaystyle \mu (\varnothing )=0.}$

Note that any sum involving ${\displaystyle +\infty }$ will equal ${\displaystyle +\infty }$, that is, ${\displaystyle a+\infty =+\infty }$ for all ${\displaystyle a}$ in the extended reals.

If the condition of non-negativity is dropped, and ${\displaystyle \mu (E)}$ only ever equals one of ${\displaystyle +\infty }$, ${\displaystyle -\infty }$, i.e. no two distinct sets have measures ${\displaystyle +\infty }$, ${\displaystyle -\infty }$, respectively, then ${\displaystyle \mu }$ is called a signed measure.

The pair ${\displaystyle (X,\Sigma )}$ is called a measurable space, and the members of ${\displaystyle \Sigma }$ are called measurable sets.

A triple ${\displaystyle (X,\Sigma ,\mu )}$ is called a measure space. A probability measure is a measure with total measure one – that is, ${\displaystyle \mu (X)=1.}$ A probability space is a measure space with a probability measure.