Layer cake representation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the layer cake representation of a non-<a href="/facts/Negative_number/LzErzCmI">negative</a>, <a href="/facts/Real_number/R02gw5Pb">real</a>-valued <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 defined on a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> 
 
 
 
 (
 Ω
 ,
 
 
 A
 
 
 ,
 μ
 )
 
 
 {\displaystyle (\Omega ,{\mathcal {A}},\mu )}
 
 is the formula

f
        (
        x
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        
          1
          
            L
            (
            f
            ,
            t
            )
          
        
        (
        x
        )
        
        
          d
        
        t
        ,
      
    
    {\displaystyle f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,\mathrm {d} t,}

for all 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
, where 
 
 
 
 
 1
 
 E
 
 
 
 
 {\displaystyle 1_{E}}
 
 denotes the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of a subset 
 
 
 
 E
 ⊆
 Ω
 
 
 {\displaystyle E\subseteq \Omega }
 
 and 
 
 
 
 L
 (
 f
 ,
 t
 )
 
 
 {\displaystyle L(f,t)}
 
 denotes the (
 
 
 
 
 
 strict
 
 
 
 
 {\displaystyle \color {red}{\text{strict}}}
 
) <a href="/facts/Level_set/JUmNbdNh">super-level set</a>:

L
        (
        f
        ,
        t
        )
        =
        {
        y
        ∈
        Ω
        ∣
        f
        (
        y
        )
        ≥
        t
        }
        
        
        
        
          
            
              or
            
            
            L
            (
            f
            ,
            t
            )
            =
            {
            y
            ∈
            Ω
            ∣
            f
            (
            y
            )
            >
            t
            }
          
        
        .
      
    
    {\displaystyle L(f,t)=\{y\in \Omega \mid f(y)\geq t\}\;\;\;{\color {red}{\text{or}}\;L(f,t)=\{y\in \Omega \mid f(y)>t\}}.}

The layer cake representation follows easily from observing that

1
          
            L
            (
            f
            ,
            t
            )
          
        
        (
        x
        )
        =
        
          1
          
            [
            0
            ,
            f
            (
            x
            )
            ]
          
        
        (
        t
        )
        
        
        
        
          
            
              or
            
            
            
              1
              
                L
                (
                f
                ,
                t
                )
              
            
            (
            x
            )
            =
            
              1
              
                [
                0
                ,
                f
                (
                x
                )
                )
              
            
            (
            t
            )
          
        
      
    
    {\displaystyle 1_{L(f,t)}(x)=1_{[0,f(x)]}(t)\;\;\;{\color {red}{\text{or}}\;1_{L(f,t)}(x)=1_{[0,f(x))}(t)}}

where either integrand gives the same integral:

f
        (
        x
        )
        =
        
          ∫
          
            0
          
          
            f
            (
            x
            )
          
        
        
        
          d
        
        t
        .
      
    
    {\displaystyle f(x)=\int _{0}^{f(x)}\,\mathrm {d} t.}

The <a href="/facts/Layer_cake/ekQUSYGb">layer cake</a> representation takes its name from the representation of the value 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 as the sum of contributions from the "layers" 
 
 
 
 L
 (
 f
 ,
 t
 )
 
 
 {\displaystyle L(f,t)}
 
: "layers"/values 
 
 
 
 t
 
 
 {\displaystyle t}
 
 below 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 contribute to the integral, while values 
 
 
 
 t
 
 
 {\displaystyle t}
 
 above 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 do not.
It is a generalization of <a href="/facts/Cavalieri%2527s_principle/CRSyxWzD">Cavalieri's principle</a> and is also known under this name.: cor. 2.2.34

Layer cake representation open-in-new

Layer cake representation