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

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...

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...

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...

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...

integration Lebesgue measure Lorentz space Lifting theory Measurable cardinal Measurable function Minkowski content Outer measure Product measure Pushforward...

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...

smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures...

For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice...

{\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)...

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...

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

("pushing forward") a measure from one measurable space to another using a measurable function. Given measurable spaces ( X 1 , Σ 1 ) {\displaystyle (X_{1}...

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...

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...

applications, for example in probability theory. Definition 1. A measurable function L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for...

{\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...

Measurable function: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function...

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...

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

\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...

an approximate limit. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory....

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

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...

. {\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...

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...

) [ y − x ] {\displaystyle f(y)\leq f(x)+f'(x)[y-x]} A Lebesgue measurable function on an interval C is concave if and only if it is midpoint concave...

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...

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...

same thing as "Lebesgue integrable" for measurable functions. The same thing goes for a complex-valued function. Let us define f + ( x ) = max ( ℜ f (...