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In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line 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 of a function f means that f is continuous on , infinitely differentiable on , and satisfies
The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on with cumulative distribution function g such that
In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on. 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.