and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure...

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...

is measurable for each element x. The concept is named after Salomon Bochner. Bochner-measurable functions are sometimes called strongly measurable, μ...

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...

not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with parameter m ...

also has κ-many measurable cardinals below it. Every measurable cardinal κ is a 0-huge cardinal because κM ⊆ M, that is, every function from κ to M is...

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...

is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes...

measurable functions on a measure space ( S , Σ , μ ) {\displaystyle (S,\Sigma ,\mu )} . Suppose that the sequence converges pointwise to a function f...

for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem...

}=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach...

Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function...

random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space. This allows consideration...

2020-12-21. Retrieved 2025-06-28. Stein, Noah. "Is Conway's base-13 function measurable?". mathoverflow. Retrieved 6 August 2023. Oman, Greg (2014). "The...

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...

\mu } -almost everywhere. In that case, the essential support of a measurable function f : X → R {\displaystyle f:X\to \mathbb {R} } written e s s s u p...

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...

{\displaystyle \nu ,} which means that there exists a ν {\displaystyle \nu } -measurable function f {\displaystyle f} taking values in [ 0 , + ∞ ) , {\displaystyle...

convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert...

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...

inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and...

be strongly measurable if and only if its restriction to each Xn is strongly measurable. This makes sense because Hx is constant on Xn. Measurable families...

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}...

function of each set C {\displaystyle C} . Further generalization is the map induced by P n {\displaystyle P_{n}} on measurable real-valued functions...

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...

(f(x)\in F(x))\,.} The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential...

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...

) {\displaystyle L^{2}(\mathbb {Q} )} denote the space of Borel-measurable functions that are square-integrable with respect to the measure Q {\displaystyle...

is a set function defined on the set of all Jordan measurable subsets of R n ; {\displaystyle \mathbb {R} ^{n};} it sends a Jordan measurable set to its...

(below a measurable cardinal of Mitchell order κ + + {\displaystyle \kappa ^{++}} ) also imply SCH. Thomas Jech: Properties of the gimel function and a classification...