Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory.
 Its statement is as follows: Let formula_1 denote the Lebesgue outer measure on formula_2, and let formula_3.
 Then formula_4 is Lebesgue measurable if and only if formula_5 for every formula_6.
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.
The intercept theorem, also known as Thales' theorem (not to be confused with another theorem with that name) or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels.
 It is equivalent to the theorem about ratios in similar triangles.
 Traditionally it is attributed to Greek mathematician Thales.
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.
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.
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue.
 It is a strong condition that depends on uniform integrability.
 It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem follows as a special case of Vitali's.
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 metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry.
 It is named after the Greek mathematician Constantin Carathéodory.
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces.
 This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem.
 The covering theorem is credited to the Italian mathematician Giuseppe Vitali.
 The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset "E"  of R by a disjoint family extracted from a "Vitali covering" of "E".
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.

Question: What Greek mathematician who spend most of his professional life in Germany lends his name to the Vitali–Carathéodory theorem?
Answer: Constantin Carathéodory