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  # Total monotonicity: Wikis
  
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  In real analysis, a branch of mathematics, Bernstein's theorem states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.
  
  Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies
  
  $(-1)^n{d^n \over dt^n} f(t) \geq 0$
  
  for all nonnegative integers n and for all t > 0. Another convention puts the opposite inequality in the above definition. Completely monotone functions are also called Bernstein functions.
  
  The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞), with cumulative distribution function g, such that
  
  $f(t) = \int_0^\infty e^{-tx} \,dg(x),$
  
  the integral being a Riemann-Stieltjes integral.
  
  Further, any Bernstein function can be written in the following form:
  
  $f(t) = a + b t +\int_0^\infty (1-e^{-t x})\mu(dx)$
  
  where $a,b \geq 0$ and μ is a measure on the positive real half-line such that
  
  $\int_0^\infty (1\wedge x) \mu(dx) <\infty.$
  
  In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0,∞). In this form it is known as the Bernstein-Widder theorem, or Hausdorff-Bernstein-Widder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.
  
  ## References
  
  • S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679.
  • D. Widder (1941). The Laplace Transform. Princeton University Press.
explanation: The provided text is an encyclopedia entry about Bernstein's theorem in real analysis. It explains the concept of total monotonicity and the theorem's statement that any totally monotone function on the half-line [0, ∞) is a mixture of exponential functions. The text also provides a detailed mathematical formulation of the theorem and its generalization, along with references to related work.
  
  This document does not contain a question or an answer to a question.
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distilabel_metadata.content: •   Wikis
  
  # Total monotonicity: Wikis
  
  Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
  
  # Encyclopedia
  
  (Redirected to Bernstein's theorem article)
  
  In real analysis, a branch of mathematics, Bernstein's theorem states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.
  
  Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies
  
  $(-1)^n{d^n \over dt^n} f(t) \geq 0$
  
  for all nonnegative integers n and for all t > 0. Another convention puts the opposite inequality in the above definition. Completely monotone functions are also called Bernstein functions.
  
  The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞), with cumulative distribution function g, such that
  
  $f(t) = \int_0^\infty e^{-tx} \,dg(x),$
  
  the integral being a Riemann-Stieltjes integral.
  
  Further, any Bernstein function can be written in the following form:
  
  $f(t) = a + b t +\int_0^\infty (1-e^{-t x})\mu(dx)$
  
  where $a,b \geq 0$ and μ is a measure on the positive real half-line such that
  
  $\int_0^\infty (1\wedge x) \mu(dx) <\infty.$
  
  In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0,∞). In this form it is known as the Bernstein-Widder theorem, or Hausdorff-Bernstein-Widder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.
  
  ## References
  
  • S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679.
  • D. Widder (1941). The Laplace Transform. Princeton University Press.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an encyclopedia entry about Bernstein's theorem in real analysis. It explains the concept of total monotonicity and the theorem's statement that any totally monotone function on the half-line [0, ∞) is a mixture of exponential functions. The text also provides a detailed mathematical formulation of the theorem and its generalization, along with references to related work.
  
  This document does not contain a question or an answer to a question.
  
  Label: no-question