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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
analysis—a 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
Weakly_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
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
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
Type of topological space
{\displaystyle L^{p}(X)} consists of (equivalence classes of) all Bochner measurable functions f {\displaystyle f} with values in the Banach space X {\displaystyle
Bochner_space
Function spaces generalizing finite-dimensional p norm spaces
{\displaystyle \{s\in S:f(s)\neq g(s)\}} is measurable and has measure zero. Similarly, a measurable function f {\displaystyle f} (and its absolute value)
Lp_space
Set theory concept
fact weakly Mahlo). All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ {\displaystyle \kappa } is measurable if
Measurable_cardinal
Association of one output to each input
logicians, give precise definitions for these weakly specified functions. These generalized functions may be critical in the development of a formalization
Function_(mathematics)
Form of continuity for functions
function is absolutely continuous. If f: [a,b] → R is absolutely continuous, then it is weakly differentiable; conversely if f: [a,b] → R is weakly differentiable
Absolute_continuity
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
Generalization of finite measure to Banach spaces Weakly measurable function James K. Brooks, Representations of weak and strong integrals in Banach spaces, Proceedings
Pettis_integral
Concept in mathematics
Vector measure – Generalization of finite measure to Banach spaces Weakly measurable function Ardent, Wolfgang; Batty, Charles J.K; Hieber, Matthias; Neubrander
Bochner_integral
Generalization of finite measure to Banach spaces
been viewed as a discrete analogue of Lyapunov's theorem. Bochner measurable function Bochner integral – Concept in mathematics Bochner space – Type of
Vector_measure
or functions is called monotone or monotonic if it is either weakly increasing x 1 ≤ x 2 ≤ ⋯ {\displaystyle x_{1}\leq x_{2}\leq \cdots } or weakly decreasing
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
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
Alexandra_Bellow
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
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
Axiom of set theory
are all pairwise congruent. The set X {\displaystyle X} will be non-measurable for any rotation-invariant countably additive measure on S {\displaystyle
Axiom_of_choice
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
Kind of large cardinal number
a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.p. 364 The hierarchy
Woodin_cardinal
Function in Boolean algebra
subsets of the Cantor space are measurable and have the property of Baire and thus that no infinite parity function exists; this holds in the Solovay
Parity_function
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
Function that ranks states of society according to their desirability
Continuity: for every profile v, the set of profiles weakly better than v and the set of profiles weakly worse than v are closed sets.[jargon] Independence
Social_welfare_function
Theorem in probability theory
theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after
Helly–Bray_theorem
Averages of repeated trials converge to the expected value
continuous at each θ ∈ Θ for almost all xs, and measurable function of x at each θ. there exists a dominating function d(x) such that E[d(X)] < ∞, and ‖ f ( x
Law_of_large_numbers
Homeomorphism between plane domains
In mathematical complex analysis, a quasiconformal mapping is a (weakly differentiable) homeomorphism between plane domains which to first order takes
Quasiconformal_mapping
inequalities. Martingales are weakly dependent [citation needed], so many results about martingales also hold true for weakly dependent sequences. An example
Weakly dependent random variables
Weakly_dependent_random_variables
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
Mathematical operator in real and harmonic analysis
jointly continuous in x and r, so the maximal function Mf, being the supremum over r > 0, is measurable. A nontrivial corollary of the Hardy–Littlewood
Hardy–Littlewood maximal function
Hardy–Littlewood_maximal_function
Real function with secant line between points above the graph itself
} This condition is only slightly weaker than convexity. For example, a real-valued Lebesgue measurable function that is midpoint-convex is convex: this
Convex_function
Stochastic process in probability theory
normal random variable N(0, P(A)(1 − P(A))) for fixed measurable set A. Similarly, for a fixed function f, G n f {\displaystyle G_{n}f} converges in distribution
Empirical_process
Type of energy
the collector) then a measurable electric current will be observed. Thermionic emission can be used to measure the work function of both the hot emitter
Work_function
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
Mathematical function characterizing set membership
In 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
Indicator_function
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
Normed vector space that is complete
the weak*-topology of the bidual. The Banach space X {\displaystyle X} is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent
Banach_space
Mathematical description of mixing substances
mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also
Mixing_(mathematics)
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
Mathematical function with no sudden changes
domains. In measure theory, a function f : E → R k {\displaystyle f:E\to \mathbb {R} ^{k}} defined on a Lebesgue measurable set E ⊆ R n {\displaystyle E\subseteq
Continuous_function
Concept in economics and decision theory
cardinalist interpretation of utility functions. Utility differences: Utility differences refer to the measurable variations in utility levels between
Utility
In mathematics, a quantitative measure of the shape of a set of points
is a sequence μ n ′ {\displaystyle {\mu _{n}}'} that weakly converges to a distribution function μ {\displaystyle \mu } having α k {\displaystyle \alpha
Moment_(mathematics)
Mathematical method
a measurable space, and F : Ω → C l ( X ) {\displaystyle F:\Omega \to \mathrm {Cl} (X)} is an F {\displaystyle {\mathcal {F}}} -weakly measurable map
Selection_theorem
Notion of convergence in mathematics
about almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere
Pointwise_convergence
Branch of mathematics
compatible with the set operations on the measurable sets. In measure theory, pointwise convergence of functions can be replaced with the notion of convergence
Mathematical_analysis
Type of vector space in math
any orthonormal sequence {fn} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {xn} is bounded, by the
Hilbert_space
Mapping function
In mathematics, an additive set function is a function μ \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets
Sigma-additive_set_function
Pseudoscientific alternative medicine practice
weakly diamagnetic (when oxygenated) or paramagnetic (when deoxygenated), the magnets used in magnetic therapy are many orders of magnitude too weak to
Magnet_therapy
Whose values lie in an infinite-dimensional vector space
The measurability of f {\displaystyle f} can be defined by a number of ways, most important of which are Bochner measurability and weak measurability. The
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Mathematical concept
Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every A ∉ I and for every function f: [κ]<ω → {0
Ramsey_cardinal
Type of topological space in mathematics
with weaker notions of locally compact. Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally
Locally_compact_space
Theory of probability
function of each set C {\displaystyle C} . Further generalization is the map induced by P n {\displaystyle P_{n}} on measurable real-valued functions
Glivenko–Cantelli_theorem
Integral expressing the amount of overlap of one function as it is shifted over another
{\displaystyle A\subset \mathbf {R} ^{d}} is a measurable set and 1 A {\displaystyle 1_{A}} is the indicator function of A {\displaystyle A} . This agrees with
Convolution
Concept within complex analysis
metric space. When 0 < p ≤ 1 {\displaystyle 0<p\leq 1} , a bounded measurable function f {\displaystyle f} of compact support is in the Hardy space H p
Hardy_space
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
Real function with finite total variation
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite):
Bounded_variation
In contrast with ordinal utility, in economics
changed. Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity. The point
Cardinal_utility
Theorem of convex functions
density function. Then Jensen's inequality becomes the following statement about convex integrals: If g is any real-valued measurable function and φ {\textstyle
Jensen's_inequality
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 {\displaystyle f:\mathbb {R} \to \mathbb {C} } is called
Fourier_transform
Property of functions which is weaker than continuity
is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively
Semi-continuity
Concept in mathematics
without the full axiom of choice, in which all sets of real numbers are measurable. Urysohn's lemma (UL) and the Tietze extension theorem (TET) are independent
Axiom_of_countable_choice
Classes of functions
{\displaystyle X_{1},\dots ,X_{n}} from P {\displaystyle P} . The class of measurable functions F {\displaystyle {\mathcal {F}}} is called a Donsker class if the
Donsker_classes
Mathematical function
(f(x)\in F(x))\,.} The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential
Choice_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)
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)
Mathematical theorem
Zbl 1156.53003. Ziemer, William P. (1989). Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics
Rademacher's_theorem
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
Superstrong cardinal Totally indescribable cardinal Weakly compact cardinal Weakly hyper-Woodin cardinal Weakly inaccessible cardinal Woodin cardinal Unfoldable
List of mathematical logic topics
List_of_mathematical_logic_topics
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
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
Locally compact topological group with an invariant averaging operation
representations are weakly contained in the left regular representation λ on L2(G). Trivial representation. The trivial representation of G is weakly contained
Amenable_group
Mathematical form
this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form
Dirichlet_form
Branch of mathematics that studies dynamical systems
a set of measure zero, where χA is the indicator function of A. The occurrence times of a measurable set A is defined as the set k1, k2, k3, ..., of times
Ergodic_theory
Functional equation
conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function f : R → R {\displaystyle
Cauchy's_functional_equation
Concept in game theory
cooperation has positive synergy, all agents (weakly) gain, and if it has negative synergy, all agents (weakly) lose. If i {\displaystyle i} and j {\displaystyle
Shapley_value
Specific figure of merit in electronics
assumption of a weakly nonlinear system, meaning that higher-order nonlinear terms are small enough to be negligible. In practice, the weakly nonlinear assumption
Third-order_intercept_point
Chemical decomposition caused by heat
experimentally; this is the temperature at which a chemical reaction occurs at a measurable rate, and is thus highly dependent on the sensitivity of the corresponding
Thermal_decomposition
Lowest possible energy of a quantum system or field
light due to quantum fluctuations should lead to this principle being measurably violated if the interactions are strong enough. Nearly all theories of
Zero-point_energy
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
Partial differential equation
measure zero, the Beltrami coefficient extends uniquely to a bounded measurable function on C. smooth on the regular set. The normalised solution of the Beltrami
Beltrami_equation
Axiomatic set theory devised by W.V.O. Quine
{\displaystyle {\mathsf {weakly\ compact}}} see weakly compact cardinal. m e a s u r a b l e {\displaystyle {\mathsf {measurable}}} see measurable cardinal. Type-style
New_Foundations
Note that f is also not measurable; an additive real function is linear if and only if it is measurable, so for every such function there is a Vitali set
Discontinuous_linear_map
Length in a vector space
number-of-non-zeros function the L 0 {\displaystyle L^{0}} norm, echoing the notation for the Lebesgue space of measurable functions. The generalization
Norm_(mathematics)
Concept in measure theory
{\displaystyle \mu } on a measurable space ( X , F ) {\displaystyle (X,{\mathcal {F}})} , a σ {\displaystyle \sigma } -finite subset is a measurable subset which is
Σ-finite_measure
Subfield of set theory
Projective hierarchy for definitions.) Actually a measurable cardinal is more than enough. A weaker principle — the existence of 0# is sufficient to prove
Determinacy
Study of optimal transportation and allocation of resources
μ k → μ {\displaystyle \mu _{k}\to \mu } weakly, a sequence ν k → ν {\displaystyle \nu _{k}\to \nu } weakly, and a sequence of optimal transport plans
Transportation theory (mathematics)
Transportation_theory_(mathematics)
Type of random mathematical object
Borel measurable sets B 1 , … , B k {\displaystyle \textstyle B_{1},\dots ,B_{k}} , an inhomogeneous Poisson process with (intensity) function λ ( x )
Poisson_point_process
Spectral density of light emitted by a black body
the specific intensity law Cλ−5e−c⁄λT where C and c denote empirically measurable constants, and where λ and T denote wavelength and temperature respectively
Planck's_law
Measure in mathematical analysis
associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence
Young_measure
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
cumulative distribution functions F 1 , F 2 , … {\displaystyle F_{1},F_{2},\ldots } , is said to converge in distribution, or converge weakly, or converge in
Convergence of random variables
Convergence_of_random_variables
Directional planes
context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are
Vertical_and_horizontal
Property of uniformly space-filling movement
) {\displaystyle (X,{\mathcal {B}})} be a measurable space. If T {\displaystyle T} is a measurable function from X {\displaystyle X} to itself and μ {\displaystyle
Ergodicity
Mathematics concept
\mathbb {R} ^{k}} , k ∈ N {\displaystyle k\in \mathbb {N} } , is any measurable function. Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb
Finite-dimensional distribution
Finite-dimensional_distribution
Type of signal in signal processing
authors use strongly white and weakly white instead. An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is
White_noise
Chemical compound
safrole, like many naturally occurring compounds, may have a small but measurable ability to induce cancer in rodents. Despite this, the effects in humans
Safrole
Collection of random variables
considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then
Stochastic_process
Cryptographic attack
library function for hashing an 8-character password into an 11-character string. On older hardware, this computation took a deliberately and measurably long
Timing_attack
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
Possible axiom for set theory
contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences
Axiom_of_determinacy
Theorem on extension of bounded linear functionals
Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set. Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox
Hahn–Banach_theorem
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
Girl/Female
Hindu
Pearl Pearly just similar to Pearl
Boy/Male
Tamil
Sukhamay | ஸூக஼மயÂ
Pleasurable
Sukhamay | ஸூக஼மயÂ
Surname or Lastname
English
English : habitational name from a place in Northamptonshire called Weekley, from Old English wīc ‘settlement’, perhaps in this case a Roman settlement, Latin vicus + lēah ‘wood’, ‘clearing’.
Boy/Male
Hindu
Immeasurable splendor
Boy/Male
Tamil
Atultejas | அதà¯à®²à®¤à¯‡à®œà®¸
Immeasurable brightness
Atultejas | அதà¯à®²à®¤à¯‡à®œà®¸
Girl/Female
Indian
Immeasurable, Boundless
Girl/Female
Tamil
Immeasurable, Boundless
Surname or Lastname
English
English : variant of Weekley.
Boy/Male
Assamese, Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Immeasurable Splendour
Boy/Male
Hindu, Indian, Marathi
Immeasurable
Boy/Male
Sikh
Pleasurable
Surname or Lastname
English
English : variant spelling of Weekley.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Immeasurable
Girl/Female
Indian
Immeasurable, Boundless
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Pleasurable
Girl/Female
Tamil
Immeasurable, Boundless
Boy/Male
Indian, Sanskrit
Pleasurable
Surname or Lastname
English
English : variant spelling of Weekley.
Boy/Male
Tamil
Aniteja | அநிதேஜ஼ாÂ
Immeasurable splendor
Aniteja | அநிதேஜ஼ாÂ
Boy/Male
Indian, Sanskrit
Immeasurable Brightness
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
Female
Hebrew
(×ֲש×ֵרָה) Hebrew name ASHERAH means "groves (for idol worship)" or "blessed, fortunate." In the bible, this is the Hebrew name for the Babylonian-Canaanite goddess Astarte. It is also the name for her images and sacred trees or poles used for worshiping her.Â
Boy/Male
Indian, Malayalam
Something Special
Female
Egyptian
, Bantanath.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Respected Brave One
Boy/Male
Indian
Arcturus brightest star in constellation bootes
Girl/Female
Indian
Someone you cannot stop loving
Boy/Male
Hindu
Boy/Male
Arabic, British, English, Muslim
Victorious; Happy; Fortunate
Girl/Female
Hindu, Indian
Strong; Soldier
Surname or Lastname
Irish
Irish : variant spelling of Noone.English, Scottish, and Dutch : from Middle English none, Middle Dutch noene ‘noon’, the time of brightest sunshine, hence perhaps nickname for a bright and cheerful person or for someone born at that time of day. The word is derived from Latin nona (hora) ‘ninth (hour)’, i.e. about three o’clock. The change in meaning of the vocabulary word from mid-afternoon to midday, probably occurred as a result of monastic meal times being brought forward.
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
WEAKLY MEASURABLE-FUNCTION
v. t.
To make weak; to lessen the strength of; to deprive of strength; to debilitate; to enfeeble; to enervate; as, to weaken the body or the mind; to weaken the hands of a magistrate; to weaken the force of an objection or an argument.
a.
Neatly; dexterously; nimbly.
adv.
In a weak manner; with little strength or vigor; feebly.
a.
Of or pertaining to a week, or week days; as, weekly labor.
a.
Coming, happening, or done once a week; hebdomadary; as, a weekly payment; a weekly gazette.
a.
Immeasurable.
adv.
Early.
a.
Immeasurable.
a.
Capable of being measured; measurable.
a.
Immeasurable.
a.
Vacant of employment; not occupied; idle; leisure; as leisurable hours.
a.
Capable of being measured; susceptible of mensuration or computation.
a.
Not censurable.
a.
Moderate; temperate; not excessive.
a.
Deserving of censure; blamable; culpable; reprehensible; as, a censurable person, or censurable conduct.
adv.
Once a week; by hebdomadal periods; as, each performs service weekly.
a.
Happening, accruing, or coming every year; annual; as, a yearly income; a yearly feast.
a.
Immeasurable.
superl.
Not strong of constitution; infirm; feeble; as, a weakly woman; a man of a weakly constitution.
a.
To make or become weak; to weaken.