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See searches and references containing MEASURABLE FUNCTION!MEASURABLE FUNCTION
Kind of mathematical function
and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure
Measurable_function
Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued
Bochner_measurable_function
weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in
Weakly_measurable_function
Method of mathematical integration
products of a measurable set with an interval. An equivalent way to introduce the Lebesgue integral is to use so-called simple functions, which generalize
Lebesgue_integral
Conditions for switching order of integration in calculus
is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli theorems
Fubini's_theorem
Strong measurability has a number of different meanings, some of which are explained below. For a function f with values in a Banach space (or Fréchet
Strongly_measurable_function
Function spaces generalizing finite-dimensional p norm spaces
of a measurable function f and its absolute value |f|:S→[0,∞] are always the same (that is, ‖f‖p=‖|f|‖p for all p) and so a measurable function belongs
Lp_space
Generalization of mass, length, area and volume
{\displaystyle X} , defining subsets of X {\displaystyle X} that are "measurable". A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the
Measure_(mathematics)
Function that attains finitely many values
For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice
Simple_function
"Pushed forward" from one measurable space to another
("pushing forward") a measure from one measurable space to another using a measurable function. Given measurable spaces ( X 1 , Σ 1 ) {\displaystyle (X_{1}
Pushforward_measure
Mathematical concept in measure theory
an approximate limit. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.
Approximately continuous function
Approximately_continuous_function
Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function
Carathéodory_function
Mathematical function that outputs real values
a function f is such that the preimage f −1(B) of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also
Real-valued_function
Function whose absolute value has a finite integral
same thing as "Lebesgue integrable" for measurable functions. The same thing goes for a complex-valued function. Let us define f+(x)=max(ℜf(x),0) f−(x)=max(−ℜf(x)
Absolutely integrable function
Absolutely_integrable_function
Function in mathematics
in probability theory and extreme value theory. Definition 1. A measurable function L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for
Slowly_varying_function
Function whose squared absolute value has finite integral
function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite
Square-integrable_function
Theorem in measure theory
measurable functions on a measure space ( S , Σ , μ ) {\displaystyle (S,\Sigma ,\mu )} . Suppose that the sequence converges pointwise to a function f
Dominated_convergence_theorem
Theorems on the convergence of bounded monotonic sequences
that says that for sequences of non-negative pointwise-increasing measurable functions 0 ≤ f 1 ( x ) ≤ f 2 ( x ) ≤ ⋯ {\displaystyle 0\leq f_{1}(x)\leq f_{2}(x)\leq
Monotone_convergence_theorem
Measurable function: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function
List_of_types_of_functions
Statement in probability theory
The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below. Remark. The lemma remains
Doob–Dynkin_lemma
Concept in probability theory
{\displaystyle (Y,{\mathcal {B}})} arbitrary measurable spaces, and let f : X → Y {\displaystyle f:X\to Y} be a measurable function. Now define κ ( d y | x ) = δ f
Markov_kernel
Theorem in measure theory
criterion states that an almost-everywhere finite function is measurable if and only if it is a continuous function on nearly all its domain. In the informal
Lusin's_theorem
Branch of functional analysis
unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R. If ξ is an element of H, then ν ξ : E ↦ ⟨ π T ( 1 E ) ξ , ξ
Borel_functional_calculus
smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures
Baire_set
Negative of a convex function
measurable function on an interval C is concave if and only if it is midpoint concave, that is, for any x and y in C f(x+y2)≥f(x)+f(y)2 If a function
Concave_function
Description of continuous random distribution
. {\displaystyle f={\frac {dX_{*}P}{d\mu }}.} That is, f is any measurable function with the property that: Pr [ X ∈ A ] = ∫ X − 1 A d P = ∫ A f d μ
Probability_density_function
Real function with secant line between points above the graph itself
real-valued Lebesgue measurable function that is midpoint-convex is convex: this is a theorem of Sierpiński. In particular, a continuous function that is midpoint
Convex_function
Variable representing a random phenomenon
random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space. This allows consideration
Random_variable
Basic object in measure theory; set and a sigma-algebra
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets
Measurable_space
Space of bounded sequences
}=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach
L-infinity
Expected value of a random variable given that certain conditions are known to occur
measurable function such that min g measurable E ( ( X − g ( Y ) ) 2 ) = E ( ( X − e X ( Y ) ) 2 ) . {\displaystyle \min _{g{\text{ measurable }}}\operatorname
Conditional_expectation
Inputs for which a function's value is non-zero
equal μ-almost everywhere. In that case, the essential support of a measurable function f:X→ℝ written esssupp(f), is defined to be the smallest closed subset
Support_(mathematics)
Type of vector space in math
real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying ∫ 0 1 | f
Hilbert_space
Category whose objects are measurable spaces and whose morphisms are measurable maps
category because the composition of two measurable maps is again measurable, and the identity function is measurable. N.B. Some authors reserve the name Meas
Category_of_measurable_spaces
Expressing a measure as an integral of another
on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples
Radon–Nikodym_theorem
Measure with complex values
measure μ {\displaystyle \mu } on a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} is a complex-valued function μ : Σ → C {\displaystyle \mu :\Sigma
Complex_measure
Theorem in measure theory
such that: the function x↦μx is Borel measurable, in the sense that x↦μx(B) is a Borel-measurable function for each Borel-measurable set B⊆Y; μx "lives
Disintegration_theorem
Concept in mathematics
mathematics, the layer cake representation of a non-negative, real-valued measurable function f {\displaystyle f} defined on a measure space ( Ω , A , μ ) {\displaystyle
Layer_cake_representation
Category whose objects are measurable spaces and whose morphisms are Markov kernels
whose objects are measurable spaces and whose morphisms are Markov kernels. It is analogous to the category of sets and functions, but where the arrows
Category_of_Markov_kernels
Average value of a random variable
a measurable function of X , {\displaystyle X,} g ( X ) , {\displaystyle g(X),} given that X {\displaystyle X} has a probability density function f (
Expected_value
{\displaystyle \Phi \colon G\times S\to S} If Φ {\displaystyle \Phi } is a measurable function from G ⊗ S {\displaystyle {\mathcal {G}}\otimes {\mathcal {S}}} to
Measurable_acting_group
Algebraic structure of set algebra
a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The
Σ-algebra
{\displaystyle f} be a real-valued measurable function. The distribution function associated with f {\displaystyle f} is the function d f : [ 0 , ∞ ) → R ∪ { ∞
Distribution function (measure theory)
Distribution_function_(measure_theory)
Property of measure-preserving dynamical systems
a measure-preserving dynamical system is ergodic if every invariant measurable set has either measure zero or full measure. Equivalently, the system
Ergodicity
Function which is integrable on its domain
f : Ω → C {\textstyle f:\Omega \to {\mathbb {C}}} be a Lebesgue measurable function. If f {\textstyle f} on Ω {\textstyle \Omega } is such that ∫ K |
Locally_integrable_function
Operation in calculus
positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines
Integral
Maximized objective function of an optimization problem
{\displaystyle u\in U[t_{0},t_{1}]} , where u {\displaystyle u} is a Lebesgue measurable function from [ t 0 , t 1 ] {\displaystyle [t_{0},t_{1}]} to some prescribed
Value_function
E ) {\displaystyle (E,{\mathcal {E}})} a measurable space. A random element with values in E is a function X: Ω→E which is ( F , E ) {\displaystyle ({\mathcal
Random_element
Mathematical description of quantum state
measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities. Wave functions can be functions of variables other
Wave_function
Model in probability theory
Y t {\displaystyle Y_{t}} is a Σ t {\displaystyle \Sigma _{t}} -measurable function; for each t {\displaystyle t} , Y t {\displaystyle Y_{t}} lies in
Martingale (probability theory)
Martingale_(probability_theory)
Lemma in measure theory
}}_{\geq 0}})} -measurable non-negative functions f n : X → [ 0 , + ∞ ] {\displaystyle f_{n}:X\to [0,+\infty ]} . Define the function f : X → [ 0 , +
Fatou's_lemma
Concept in mathematics
measure space, and B {\displaystyle B} be a Banach space, and define a measurable function f : X → B {\displaystyle f:X\to B} . When B = R {\displaystyle B=\mathbb
Bochner_integral
Technique in integral evaluation
Borel measurable function g on Y. In geometric measure theory, integration by substitution is used with Lipschitz functions. A bi-Lipschitz function is a
Integration_by_substitution
Theorem concerning uniform convergence
for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem
Egorov's_theorem
Type of probability space
called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions. The theory of standard probability spaces
Standard_probability_space
Inequality between integrals in Lp spaces
p,q\in [1,\infty ]} with 1/p + 1/q = 1. Then for all measurable real- or complex-valued functions f and g on S, ‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖ q . {\displaystyle
Hölder's_inequality
Topics referred to by the same term
Pushforward measure, measure induced on the target measure space by a measurable function Pushout (category theory), the categorical dual of pullback Direct
Pushforward
one with the fewest measurable sets) such that every continuous linear function on X {\displaystyle X} is a measurable function. In general, A ( X ,
Cylindrical_σ-algebra
Form of continuity for functions
{\displaystyle \nu ,} which means that there exists a ν {\displaystyle \nu } -measurable function f {\displaystyle f} taking values in [ 0 , + ∞ ) , {\displaystyle
Absolute_continuity
Type of random mathematical object
collection of disjoint bounded Borel measurable sets B1,…,Bk, an inhomogeneous Poisson process with (intensity) function λ(x) has the finite-dimensional distribution:
Poisson_point_process
Mathematical functions
{\displaystyle X} , and θ {\displaystyle \theta } a Young function. For any measurable function f {\displaystyle f} on X {\displaystyle X} , we define the
Young_function
Theorem on operator interpolation
prove the result for simple functions and eventually show how the argument can be extended by density to all measurable functions. By symmetry, let us assume
Riesz–Thorin_theorem
Mathematical function for the probability a given outcome occurs in an experiment
measurable function X {\displaystyle X} from a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} to a measurable space
Probability_distribution
Stochastic way of assigning quantities across a space
f(x)\;\operatorname {E} \zeta (\mathrm {d} x)} for every positive measurable function f {\displaystyle f} is called the intensity measure of ζ {\displaystyle
Random_measure
Theorem in probability and statistics
measure space (Ω, μ) and a measurable map X from Ω to a measurable space Ω'. The theorem then says that for any measurable function g on Ω' which is valued
Law of the unconscious statistician
Law_of_the_unconscious_statistician
Association of one output to each input
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
Function_(mathematics)
Inequalities in measure theory
Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually. Let X {\displaystyle X} be
Clarkson's_inequalities
Set theory concept
In mathematics, specifically in set theory, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces
Measurable_cardinal
Set of functions between two fixed sets
≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } , is the Lp space of measurable functions whose p-norm ‖ f ‖ p = ( ∫ Ω | f | p ) 1 / p {\textstyle \|f\|_{p}=\left(\int
Function_space
Set with no meaningful size
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful size, such as a length, area, or volume. The term is normally applied
Non-measurable_set
of a measurable function both to higher-dimensional domains and, in the case of one dimension, to a superset of the Lebesgue integrable functions. UC Davis
Washek_Pfeffer
Mathematical theorem about Dini derivatives
a function that hold almost everywhere. Denjoy (1915) proved the theorem for continuous functions, Young (1917) extended it to measurable functions, and
Denjoy–Young–Saks_theorem
Construct related to weighted sums and averages
\Omega \to \mathbb {R} ^{+}} is a non-negative measurable function. In this context, the weight function w ( x ) {\displaystyle w(x)} is sometimes referred
Weight_function
Broadest definition of sizes in integer-dimensional spaces
axioms of set theory: the sets to which it applies are called Lebesgue-measurable. One way to characterize the Lebesgue measure is to first define it on
Lebesgue_measure
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Measure of prediction accuracy of a forecast
Regression models aim at finding a good model for the pair, that is a measurable function g from R d {\displaystyle \mathbb {R} ^{d}} to R {\displaystyle \mathbb
Mean absolute percentage error
Mean_absolute_percentage_error
Function type in graph theory
statistics, a graphon (also known as a graph limit) is a symmetric measurable function W : [ 0 , 1 ] 2 → [ 0 , 1 ] {\displaystyle W:[0,1]^{2}\to [0,1]}
Graphon
Mathematics of real numbers and real functions
spaces, with the usual identification of measurable functions that are equal almost everywhere. These function spaces, while all being infinite dimensional
Real_analysis
Differential equations involving stochastic processes
{\displaystyle F:\Omega \to U} be some initial condition, meaning it is a measurable function with respect to the initial σ-algebra. Let ζ : Ω → R ¯ + {\displaystyle
Stochastic differential equation
Stochastic_differential_equation
Theorem of convex functions
space of T: E|⟨z,X⟩|<∞, and ⟨z,E[X]⟩=E[⟨z,X⟩]. Then, for any measurable convex function φ and any sub-σ-algebra 𝔊 of 𝔉: φ(E[X∣𝔊])≤E[φ(X)∣𝔊]. Here
Jensen's_inequality
Mathematical inequality about the convolution of two functions
any f ∈ L p ( G , μ ) {\displaystyle f\in L^{p}(G,\mu )} and any measurable function g {\displaystyle g} on G {\displaystyle G} that belongs to the weak
Young's convolution inequality
Young's_convolution_inequality
Measure used in functional analysis
to all bounded complex-valued measurable functions on X, and we have the following. Theorem—For any bounded Borel function f {\displaystyle f} on X {\displaystyle
Projection-valued_measure
Type of statistical analysis
regression function) belongs to a set of functions parametrized by a set Θ {\displaystyle \Theta } , one searches for a (measurable) function T n : X n
Nonparametric_statistics
Romanian-American mathematician (1935–2025)
lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly measurable function; this gives a one
Alexandra_Bellow
measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection
Kuratowski and Ryll-Nardzewski measurable selection theorem
Kuratowski_and_Ryll-Nardzewski_measurable_selection_theorem
Stochastic diffusion process in probability theory
of the pool will only depend on the particle itself. Consider a measurable function σ:ℝd×𝒫(ℝd)→ℳd(ℝ) where 𝒫(ℝd) is the space of probability distributions
McKean–Vlasov_process
Area of mathematics
{\displaystyle [\,f\,]} of measurable functions whose absolute value's p {\displaystyle p} -th power has finite integral; that is, functions f {\displaystyle f}
Functional_analysis
Random measure in probability theory
is simply the empirical mean of the indicator function, Pn(A) = Pn IA. For a fixed measurable function f {\displaystyle f} , P n f {\displaystyle P_{n}f}
Empirical_measure
Mathematical concept
every n > N and for every measurable set A. As before, this implies convergence of integrals against bounded measurable functions, but this time convergence
Convergence_of_measures
Mathematical transform that expresses a function of time as a function of frequency
forward and the reverse transform. The signs must be opposites. A measurable function f:ℝ→ℂ is called (Lebesgue) integrable if the Lebesgue integral of
Fourier_transform
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
Notion in measure theory
picks from every equivalence class [f] of bounded measurable functions modulo negligible functions a representative— which is henceforth written T([f])
Lifting_theory
Machine learning framework
μD(x)=δD(x) for a measurable function D:Ω→[0,1], in which case L(μG,μD):=Ex∼μref[lnD(x)]+Ex∼μG[ln(1−D(x))]. To define suitable density functions, we define
Generative adversarial network
Generative_adversarial_network
Mathematical set with some added structure
of sets (or functions) irrespective of any topology. Every subset of a measurable space is itself a measurable space. Standard measurable spaces (also
Space_(mathematics)
application of the theorem is that, given any measurable function g:Y→X, there exists a universally measurable function f:g(Y)⊂X→Y such that g(f(x))=x for all
Jankov–von Neumann uniformization theorem
Jankov–von_Neumann_uniformization_theorem
function ƒ ∈ W1,1 loc(Ω,Rn) has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO
Distortion_(mathematics)
\Omega )} be a measurable space and let h : X → [ 0 , 1 ] {\displaystyle h:X\to [0,1]} be an Ω {\displaystyle \Omega } -measurable function. The Sugeno integral
Sugeno_integral
an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently
Markov_operator
Definition of mathematical integration
])}{2\varepsilon }}=1} (where μ denotes Lebesgue measure). A Lebesgue-measurable function g : E → R is said to have approximate limit y at x (a point of density
Khinchin_integral
Inequality in mathematics
integral version of Hardy's inequality states the following: if f is a measurable function with non-negative values, then ∫ 0 ∞ ( 1 x ∫ 0 x f ( t ) d t ) p
Hardy's_inequality
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MEASURABLE FUNCTION
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MEASURABLE FUNCTION
MEASURABLE FUNCTION
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