Is there a measure $\nu$ on $[0,\infty)$ such that $$ \ln x=\int_{0}^{\infty}d\nu\left(y\right)/\left(x+y-1\right)? $$ Thanks for any helpful answers!
Is there a measure
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2A signed measure? – 2012-10-10
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0Dear copper.hat, it should be a non-negative measure. Thanks. – 2012-10-10
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0If $\nu \leq 0$, then $\ln 1 = 0 = \int_0^\infty \frac{d \nu (y)}{y}$ would give $\nu = 0$ on $[0, \infty)$? – 2012-10-10
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0Oops, it should be a non-negative measure. Thanks. – 2012-10-10
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2Hmm, if $\nu$ is non-negative, the same reasoning would give $v = 0$ as in the above case. (I don't know if a suitable signed measure exists, but a non-negative or non-negative one does not exist.) – 2012-10-10
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0For which $x$ such an equation has to be true? – 2012-10-11