Bernstein's theorem on monotone functions

In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line [0, ∞) that is completely monotone is a mixture of exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on [0, ∞). In one important special case the mixture is a weighted average, or expected value. It is also known as the Bernstein–Widder theorem or Hausdorff–Bernstein–Widder theorem.

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