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
$$ W:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R}\cup\left\{ +\infty\right\} $$, for $$ \Omega\subseteq\mathbb{R}^{d} $$ endowed with the Lebesgue measure, is a Carathéodory function if:

1. The mapping $$ x\mapsto W\left(x,\xi\right) $$ is Lebesgue-measurable for every $$ \xi\in\mathbb{R}^{N} $$.

2. the mapping $$ \xi\mapsto W\left(x,\xi\right) $$ is continuous for almost every $$ x\in\Omega $$.

The main merit of Carathéodory function is the following: If $$ W:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R} $$ is a Carathéodory function and $$ u:\Omega\rightarrow\mathbb{R}^{N} $$ is Lebesgue-measurable, then the composition $$ x\mapsto W\left(x,u\left(x\right)\right) $$ is Lebesgue-measurable.

Example
Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional $$ \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R}\cup\left\{ +\infty\right\} $$ where $$ W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) $$ is the Sobolev space, the space consisting of all function $$ u:\Omega\rightarrow\mathbb{R}^{m} $$ that are weakly differentiable and that the function itself and all its first order derivative are in $$ L^{p}\left(\Omega;\mathbb{R}^{m}\right) $$; and where $$ \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx $$ for some $$ W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} $$, a Carathéodory function. The fact that $$ W $$ is a Carathéodory function ensures us that $$ \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx $$ is well-defined.

p-growth
If $$ W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} $$ is Carathéodory and satisfies $$ \left|W\left(x,v,A\right)\right|\leq C\left(1+\left|v\right|^{p}+\left|A\right|^{p}\right) $$ for some $$ C>0 $$ (this condition is called "p-growth"), then $$ \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R} $$ where $$ \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx $$ is finite, and continuous in the strong topology (i.e. in the norm) of $$ W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) $$.