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  "body": "\n## Statement\n\nLet $f \\colon [0, \\infty) \\to \\R$ be locally integrable with $f(t) \\ge 0$ for all $t$.\nAssume that its [[Laplace transform|05XT5AVJBK]] $F(s) = \\int_0^\\infty f(t) e^{-st}\\, dt$ is finite for every $s \u003e 0$.\nLet $c \\ge 0$.\nIf $s F(s) \\to c$ as $s \\to 0^+$, then\n$$\\frac{1}{T} \\int_0^T f(t)\\, dt \\to c \\quad (T \\to \\infty).$$\n\n## Remarks\n\nThis is the Hardy–Littlewood Tauberian theorem for the Laplace transform; the condition $f \\ge 0$ is the [[Tauberian condition|6ZDP261K7R]].\nFor $c \u003e 0$ it is the case $\\rho = 1$ of [[Karamata's Tauberian theorem|DBEGF38V9H]], applied to $U(x) = \\int_0^x f(t)\\, dt$.\nThe proof follows Karamata.\nOne shows $s\\int_0^\\infty f(t) e^{-st} P(e^{-st})\\, dt \\to c \\int_0^1 P(x)\\, dx$ for polynomials $P$.\nThen one approximates an indicator function by polynomials from above and below, using $f \\ge 0$.\nThe Tauberian condition cannot be dropped.\nLet $a_n = 1 + (-1)^n n$ and let $f = a_n$ on $[n, n + 1)$; then $sF(s) = (1 - x)\\sum_n a_n x^n$ with $x = e^{-s}$, which tends to $1$ as $s \\to 0^+$, as in the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]] for series.\nBut $\\frac{1}{N} \\int_0^N f = \\frac{1}{N} \\sum_{k \u003c N} a_k$ is close to $3/2$ for odd $N$ and to $1/2$ for even $N$, so it has no limit.\nConverse direction: the Abelian theorem gives $sF(s) \\to c$ from the convergence of the averages, with no sign condition (see [[Abelian theorems for integral transforms|16W9PP8ET0]]).\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform smooths a function out, so knowing how the transform behaves does not, in general, tell how the function behaves.\nA Tauberian theorem says that it does, provided the function cannot oscillate wildly.\nHere the extra condition is that $f$ is never negative: then a limit of $sF(s)$ at $0$ forces the long-run average of $f$ to have the same limit.\n\n### L2 human\n\nTauberian theorems are partial converses of Abelian theorems, and each needs a side condition such as positivity, monotonicity or a growth bound.\nKaramata's theorem extends the statement to regularly varying growth, with a factor $\\Gamma(1 + \\rho)$.\nWiener's Tauberian theorem treats general convolution kernels.\nThe Wiener–Ikehara theorem, a Tauberian theorem for the Laplace transform with a condition on the boundary line, gives a short proof of the prime number theorem.\nIn probability, this theorem relates the behaviour of a Laplace transform near $0$ to the tail of a distribution.\n",
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