Lebesgue measureIn mathematics, Lebesgue measure is the standard way of assigning a notion of length to subsets of the real line, area to regions of the Euclidean plane, and volume to subsets of Euclidean space in dimensions three and higher.Support (mathematics)In mathematics, the support of a real-valuedfunctionf{\displaystyle f} is the subset of the function's domain consisting of those elements that are not mapped to zero. If the domain of f{\displaystyle f} is a topological space, then the support of f{\displaystyle f} is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.
Ring of setsIn mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of setsR{\displaystyle {\mathcal {R}}} is called a ring (of sets) if it is closed under union and intersection.
Lp spaceIn mathematics, the L spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.
Indicator functionIn mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then the indicator function of A is the function 1 A {\displaystyle \mathbf {1} _{A}} defined by 1 A ( x ) = 1 {\displaystyle \mathbf {1} _{A}\!(x)=1} if x ∈ A , {\displaystyle x\in A,} and 1 A ( x ) = 0 {\displaystyle \mathbf {1} _{A}\!(x)=0} otherwise.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined.Function spaceIn mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Compact spaceIn mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle.
Partial functionIn mathematics, a partial functionf from a setX to a set Y is a function from a subsetS of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.