Bernstein's theorem on monotone functions

In <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Bernstein's theorem states that every <a href="/facts/Real_number/R02gw5Pb">real</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> on the half-line [0, ∞) that is totally monotone is a mixture of <a href="/facts/Exponential_function/ewlYxvR2">exponential functions</a>. In one important special case the mixture is a <a href="/facts/Weighted_average/VHyJfPxb">weighted average</a>, or <a href="/facts/Expected_value/1XV0JKL8">expected value</a>.
Total monotonicity (sometimes also complete monotonicity) of a function f means that f is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> on [0, ∞), <a href="/facts/Infinitely_differentiable/mffL4ch2">infinitely differentiable</a> on (0, ∞), and satisfies

(
        −
        1
        
          )
          
            n
          
        
        
          
            
              d
              
                n
              
            
            
              d
              
                t
                
                  n
                
              
            
          
        
        f
        (
        t
        )
        ≥
        0
      
    
    {\displaystyle (-1)^{n}{\frac {d^{n}}{dt^{n}}}f(t)\geq 0}

for all nonnegative <a href="/facts/Integer/PUwwrawx">integers</a> n and for all t > 0. Another convention puts the opposite <a href="/facts/Inequality_(mathematics)/qkze4zdI">inequality</a> in the above definition. 
The "weighted average" statement can be characterized thus: there is a non-negative finite <a href="/facts/Borel_measure/8cYR8Ysw">Borel measure</a> on [0, ∞) with <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> g such that

f
        (
        t
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        
          e
          
            −
            t
            x
          
        
        
        d
        g
        (
        x
        )
        ,
      
    
    {\displaystyle f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x),}

the <a href="/facts/Integral/V7lyV997">integral</a> being a <a href="/facts/Riemann%25E2%2580%2593Stieltjes_integral/C08wyfWn">Riemann–Stieltjes integral</a>.
In more abstract language, the theorem characterises <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transforms</a> of positive <a href="/facts/Borel_measure/8cYR8Ysw">Borel measures</a> on [0, ∞). In this form it is known as the Bernstein–Widder theorem, or Hausdorff–Bernstein–Widder theorem. <a href="/facts/Felix_Hausdorff/Rn6DuY2x">Felix Hausdorff</a> had earlier characterised <a href="/facts/Complete_monotonicity/1LdQ8rur">completely monotone sequences</a>. These are the sequences occurring in the <a href="/facts/Hausdorff_moment_problem/1LdQ8rur">Hausdorff moment problem</a>.

Bernstein's theorem on monotone functions open-in-new

Bernstein's theorem on monotone functions