Carathéodory function
In mathematical analysis, a Carathéodory function (or  Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory. Definition 

, for  endowed with the Lebesgue measure, is a Carathéodory function if:

1. The mapping  is Lebesgue-measurable for every . 2. the mapping  is continuous for almost every . The main merit of Carathéodory function is the following: If  is a Carathéodory  function and  is Lebesgue-measurable, then the composition  is Lebesgue-measurable. Example 
Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional  where  is the Sobolev space, the space consisting of all function  that are weakly differentiable and that the function itself and all its first order derivative are in ; and where  for some , a Carathéodory function. The fact that  is a Carathéodory function ensures us that  is well-defined. p-growth 

If  is Carathéodory and satisfies  for some  (this condition is called "p-growth"), then  where  is finite, and continuous in the strong topology (i.e. in the norm) of . References 

Calculus of variations