measure theory 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 (mathematics), magnitude, mass, and probability of events. These seemingl ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a continuity set of a measure is any

Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...

such that

\mu(\partial B) = 0,

where

\partial B

is the (topological) boundary of . For

signed measure In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign. Definition There are two slightly different concepts of a signed measure, de ...

s, one instead asks that

, \mu, (\partial B) = 0.

The collection of all continuity sets for a given measure forms a

ring of sets (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

.Cuppens, R. (1975) Decomposition of multivariate probability. Academic Press, New York. Similarly, for a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

, a set is called a continuity set of if

\Pr \in \partial B = 0.

Continuity set of a function

The continuity set of a function is the set of points where is continuous.

References

Measure theory {{mathanalysis-stub