In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given...

In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory: Carathéodory's theorem (conformal mapping)...

measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a...

Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set...

lower than the one provided by Carathéodory's theorem can be obtained. He is credited with the authorship of the Carathéodory conjecture claiming that a closed...

Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem,...

Extension theorem may refer to: Carathéodory's extension theorem - a theorem in measure theory, named after the Greek mathematician Constantin Carathéodory...

of them are nonzero. With the lemma, Carathéodory's theorem is a simple extension: Proof of Carathéodory's theorem For any x ∈ C o n v ( S ) {\displaystyle...

The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A × B) = μ1(A)μ2(B)...

Brunn–Minkowski theorem (Riemannian geometry) Cameron–Martin theorem (measure theory) Carathéodory's theorem (measure theory) Carathéodory's extension theorem (measure...

to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure...

Assouad–Nagata dimension n is n-dimensional "at every scale". Carathéodory's extension theorem Geometric set cover problem Dimension theory Metacompact space...

construction of the Lebesgue measure is an application of Carathéodory's extension theorem. It proceeds as follows. Fix n ∈ N {\displaystyle n\in \mathbb...

-algebra generated by A {\displaystyle {\mathcal {A}}} . Then by Carathéodory's extension theorem the measure μ {\displaystyle \mu } on A {\displaystyle {\mathcal...

\mu _{1}=\mu _{2}.} This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If this result does not seem very remarkable...

this τ is defined on the power set of all subsets of X. By Carathéodory's extension theorem, the outer measure can be promoted to a full measure; the associated...

deduced by approximating by polygons. The theorem is also an immediate consequence of Carathéodory's extension theorem for conformal mappings, as discussed...

it is a metric outer measure). By Carathéodory's extension theorem, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is...

the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem) does not make it obvious whether non-measurable sets exist...

the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential...

proof of Carathéodory's theorem uses a technique of examining solutions to systems of linear equations, similar to the proof of Radon's theorem, to eliminate...

common. Carathéodory's theorem Doignon's theorem Kirchberger's theorem Shapley–Folkman lemma Krein–Milman theorem Choquet theory Radon's theorem, and its...

{\displaystyle {\mathcal {F}}} . For technical details see Carathéodory's extension theorem. Sets belonging to F {\displaystyle {\mathcal {F}}} are called...

polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial...

Set cover problem Vertex cover Lebesgue covering dimension Carathéodory's extension theorem Fowler, R.J.; Paterson, M.S.; Tanimoto, S.L. (1981), "Optimal...

von Neumann algebra Almost everywhere Carathéodory's extension theorem Content (measure theory) Fubini's theorem Fatou's lemma Fuzzy measure theory Geometric...

domain. Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets a signed measure...

left-continuous, w([s,t)) = g(t) − g(s) and w({b}) = 0). By Carathéodory's extension theorem, there is a unique Borel measure μg on [a, b] which agrees...

spaces Lévy–Steinitz theorem on the convergence of rearranged infinite series of vectors Several variations of Carathéodory's theorem (convex hull) on the...

analysis. The Phragmén–Lindelöf principle, an extension to unbounded domains. The Borel–Carathéodory theorem, which bounds an analytic function in terms...