List of continuity-related mathematical topics

Last updated

In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways.

Contents

Continuity of functions and measures

Continuum

See also

Related Research Articles

<span class="mw-page-title-main">Expected value</span> The average value of a random variable

In probability theory, the expected value is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable.

In mathematical analysis, a null set is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.

<span class="mw-page-title-main">Probability distribution</span> Mathematical function for the probability a given outcome occurs in an experiment

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.

<span class="mw-page-title-main">Random variable</span> Variable representing a random phenomenon

A random variable is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes in a sample space to a measurable space, often the real numbers.

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematical analysis and in probability theory, a σ-algebra on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The pair is called a measurable space.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

<span class="mw-page-title-main">Probability mass function</span> Discrete-variable probability distribution

In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line:

In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

In mathematics, a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by or .

In mathematics, a subset of a Polish space is universally measurable if it is measurable with respect to every complete probability measure on that measures all Borel subsets of . In particular, a universally measurable set of reals is necessarily Lebesgue measurable.

In mathematics, two positive measures and defined on a measurable space are called singular if there exist two disjoint sets whose union is such that is zero on all measurable subsets of while is zero on all measurable subsets of This is denoted by

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures and such that:

In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

<span class="mw-page-title-main">Lebesgue integration</span> Method of integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

This is a glossary of set theory.