In measure theory, Carathéodory's extension theorem (named after the Greek mathematician Constantin Carathéodory) states that any measure defined on a given ring "R" of subsets of a given set "Ω" can be extended to the σ-algebra generated by "R", and this extension is unique if the measure is σ-finite.
 Consequently, any measure on a space containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers.
 This is an extremely powerful result of measure theory, and proves, for example, the existence of the Lebesgue measure.
In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L from above and below by lower- and upper-semicontinuous functions, respectively.
 It is named after Giuseppe Vitali and Constantin Carathéodory.
In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912.
 The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions.
 The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.
In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set.
 The idea is that one can extend a complex-analytic function (from here on called simply "analytic function") along curves starting in the original domain of the function and ending in the larger set.
 A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set.
 The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued.
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem.
 The theorem, first proved in , states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve.
 The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.
In mathematics, a Borel isomorphism is a measurable bijective function between two measurable standard Borel spaces.
 By Souslin's theorem in standard Borel spaces (a set that is both analytic and coanalytic is necessarily Borel), the inverse of any such measurable bijective function is also measurable.
 Borel isomorphisms are closed under composition and under taking of inverses.
 The set of Borel isomorphisms clearly forms a group under composition.
 Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part.
 It is an application of the maximum modulus principle.
 It is named for Émile Borel and Constantin Carathéodory.
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function.
 This article will provide a brief review of growth rates, including the idea of a function of exponential type.
 Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated.
 In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions.
 It is a generalization of Peano's existence theorem.
 Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations.
 The theorem is named after Constantin Carathéodory.
Constantin Carathéodory (Greek: Κωνσταντίνος Καραθεοδωρή "Konstantinos Karatheodori"; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany.
 He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory.
 His work also includes important results in conformal representations and in the theory of boundary correspondence.
 In 1909, Carathéodory pioneered the "Axiomatic Formulation of Thermodynamics" along a purely geometrical approach.

Question: In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part, it is named for Émile Borel and Constantin Carathéodory, was a Greek mathematician who spent most of his professional career in which country?
Answer: Germany