Countably Additive Function

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as 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.

Definition

Let $X$ be a set and $\Sigma$ a $\sigma$-algebra over $X.$ A set function $\mu$ from $\Sigma$ to the extended real number line is called a measure if it satisfies the following properties:

*Non-negativity: For all $E$ in $\Sigma,$ we have $\mu(E) \geq 0.$
*Null empty set: $\mu(\varnothing) = 0.$
*Countable additivity (or $\sigma$-additivity): For all countable collections $\_^\infty$ of pairwise disjoint sets in Σ,$\mu\left(\bigcup_^\infty E_k\right)=\sum_^\infty \mu(E_k).$

If at least one set $E$ has finite measure, then the requirement that $\mu(\varnothing) = 0$ is met automatically. Indeed, by countable additivity,
$\mu(E)=\mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing),$
and therefore $\mu(\varnothing)=0.$

If the condition of non-negativity is omitted but the second and third of these conditions are met, and $\mu$ takes on at most one of the values $\pm \infty,$ then $\mu$ is called a ''signed measure''.

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

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

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.

Instances

Some important measures are listed here.

* The