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.
 
  
    
      
        W
        :
        Ω
        ×
        
          
            R
          
          
            N
          
        
        →
        
          R
        
        ∪
        
          {
          
            +
            ∞
          
          }
        
      
    
    {\displaystyle W:\Omega \times \mathbb {R} ^{N}\rightarrow \mathbb {R} \cup \left\{+\infty \right\}}
  
, for 
  
    
      
        Ω
        ⊆
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \Omega \subseteq \mathbb {R} ^{d}}
  
 endowed with the Lebesgue measure, is a Carathéodory function if:
 1. The mapping 
  
    
      
        x
        ↦
        W
        
          (
          
            x
            ,
            ξ
          
          )
        
      
    
    {\displaystyle x\mapsto W\left(x,\xi \right)}
  
 is Lebesgue-measurable for every 
  
    
      
        ξ
        ∈
        
          
            R
          
          
            N
          
        
      
    
    {\displaystyle \xi \in \mathbb {R} ^{N}}
  
.
 2. the mapping 
  
    
      
        ξ
        ↦
        W
        
          (
          
            x
            ,
            ξ
          
          )
        
      
    
    {\displaystyle \xi \mapsto W\left(x,\xi \right)}
  
 is continuous for almost every 
  
    
      
        x
        ∈
        Ω
      
    
    {\displaystyle x\in \Omega }
  
. 
 The main merit of Carathéodory function is the following: If 
  
    
      
        W
        :
        Ω
        ×
        
          
            R
          
          
            N
          
        
        →
        
          R
        
      
    
    {\displaystyle W:\Omega \times \mathbb {R} ^{N}\rightarrow \mathbb {R} }
  
 is a Carathéodory  function and 
  
    
      
        u
        :
        Ω
        →
        
          
            R
          
          
            N
          
        
      
    
    {\displaystyle u:\Omega \rightarrow \mathbb {R} ^{N}}
  
 is Lebesgue-measurable, then the composition 
  
    
      
        x
        ↦
        W
        
          (
          
            x
            ,
            u
            
              (
              x
              )
            
          
          )
        
      
    
    {\displaystyle x\mapsto W\left(x,u\left(x\right)\right)}
  
 is Lebesgue-measurable. 
 Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional 
  
    
      
        
          
            F
          
        
        :
        
          W
          
            1
            ,
            p
          
        
        
          (
          
            Ω
            ;
            
              
                R
              
              
                m
              
            
          
          )
        
        →
        
          R
        
        ∪
        
          {
          
            +
            ∞
          
          }
        
      
    
    {\displaystyle {\mathcal {F}}:W^{1,p}\left(\Omega ;\mathbb {R} ^{m}\right)\rightarrow \mathbb {R} \cup \left\{+\infty \right\}}
  
 where 
  
    
      
        
          W
          
            1
            ,
            p
          
        
        
          (
          
            Ω
            ;
            
              
                R
              
              
                m
              
            
          
          )
        
      
    
    {\displaystyle W^{1,p}\left(\Omega ;\mathbb {R} ^{m}\right)}
  
 is the Sobolev space, the space consisting of all function 
  
    
      
        u
        :
        Ω
        →
        
          
            R
          
          
            m
          
        
      
    
    {\displaystyle 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
          
        
        
          (
          
            Ω
            ;
            
              
                R
              
              
                m
              
            
          
          )
        
      
    
    {\displaystyle L^{p}\left(\Omega ;\mathbb {R} ^{m}\right)}
  
; and where 
  
    
      
        
          
            F
          
        
        
          [
          u
          ]
        
        =
        
          ∫
          
            Ω
          
        
        W
        
          (
          
            x
            ,
            u
            
              (
              x
              )
            
            ,
            ∇
            u
            
              (
              x
              )
            
          
          )
        
        d
        x
      
    
    {\displaystyle {\mathcal {F}}\left[u\right]=\int _{\Omega }W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx}
  
 for some 
  
    
      
        W
        :
        Ω
        ×
        
          
            R
          
          
            m
          
        
        ×
        
          
            R
          
          
            d
            ×
            m
          
        
        →
        
          R
        
      
    
    {\displaystyle W:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{d\times m}\rightarrow \mathbb {R} }
  
, a Carathéodory function.
The fact that 
  
    
      
        W
      
    
    {\displaystyle W}
  
 is a Carathéodory function ensures us that 
  
    
      
        
          
            F
          
        
        
          [
          u
          ]
        
        =
        
          ∫
          
            Ω
          
        
        W
        
          (
          
            x
            ,
            u
            
              (
              x
              )
            
            ,
            ∇
            u
            
              (
              x
              )
            
          
          )
        
        d
        x
      
    
    {\displaystyle {\mathcal {F}}\left[u\right]=\int _{\Omega }W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx}
  
 is well-defined.
 If 
  
    
      
        W
        :
        Ω
        ×
        
          
            R
          
          
            m
          
        
        ×
        
          
            R
          
          
            d
            ×
            m
          
        
        →
        
          R
        
      
    
    {\displaystyle W:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{d\times m}\rightarrow \mathbb {R} }
  
 is Carathéodory and satisfies 
  
    
      
        
          |
          
            W
            
              (
              
                x
                ,
                v
                ,
                A
              
              )
            
          
          |
        
        ≤
        C
        
          (
          
            1
            +
            
              
                |
                v
                |
              
              
                p
              
            
            +
            
              
                |
                A
                |
              
              
                p
              
            
          
          )
        
      
    
    {\displaystyle \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
      
    
    {\displaystyle C>0}
  
 (this condition is called "p-growth"), then 
  
    
      
        
          
            F
          
        
        :
        
          W
          
            1
            ,
            p
          
        
        
          (
          
            Ω
            ;
            
              
                R
              
              
                m
              
            
          
          )
        
        →
        
          R
        
      
    
    {\displaystyle {\mathcal {F}}:W^{1,p}\left(\Omega ;\mathbb {R} ^{m}\right)\rightarrow \mathbb {R} }
  
 where 
  
    
      
        
          
            F
          
        
        
          [
          u
          ]
        
        =
        
          ∫
          
            Ω
          
        
        W
        
          (
          
            x
            ,
            u
            
              (
              x
              )
            
            ,
            ∇
            u
            
              (
              x
              )
            
          
          )
        
        d
        x
      
    
    {\displaystyle {\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
          
        
        
          (
          
            Ω
            ;
            
              
                R
              
              
                m
              
            
          
          )
        
      
    
    {\displaystyle W^{1,p}\left(\Omega ;\mathbb {R} ^{m}\right)}
  
.