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In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line Total monotonicity (sometimes also ''complete monotonicity'') of a function ''f'' means that ''f'' is continuous on : for all nonnegative integers ''n'' and for all ''t'' > 0. Another convention puts the opposite inequality in the above definition. The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on : the integral being a Riemann–Stieltjes integral. Nonnegative functions whose derivative is completely monotone are called ''Bernstein functions''. Every Bernstein function has the Lévy-Khintchine representation: : where and is a measure on the positive real half-line such that : 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== * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bernstein's theorem on monotone functions」の詳細全文を読む スポンサード リンク
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