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Dear Mathoverflow,

I would like to know if the nomenclature of mathematics has a name for Radon-Nikodym derivatives that are bounded away from zero and infinity almost everywhere. As in for equivalent measures $\mu, \nu$, there exists constants $c,C$ such that $$ 0 < c \leq \frac{d\nu}{d\mu}(x) \leq C < \infty$$ for $\mu$-almost every $x$.

Such measures could be called boundedly equivalent. But if there already exists a name, I'd like to use it.

Another possibility is to say the measures are correlated. Intuitively the condition means $\mu$ and $\nu$ either both give small or large values to the same $x$. However this is word already has a lot of meaning in maths - perhaps it is best not to add more.

Also, I'm open to suggestion if someone would like to offer a better name.

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1 Answer 1

up vote 4 down vote accepted

Good question. I am not aware of any standard terminology, although this condition is quite natural and appears pretty often. I would rather call such measures uniformly equivalent. As for "correlated" - as you say, it would create wrong connotations.

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The measures induce (semi-)norms $\| f\|_\nu = \int |f|d\nu$ and $\|f\|_\mu$ which are then equivalent in the usual notation for normed spaces, i.e. $c\|f\|_\mu \le \|f\|_\nu \le C\|f\|_\mu$. Unfortunately, equivalence of measures is used in a different sense and therefore I would also vote for uniform equivalence. –  Jochen Wengenroth Jul 24 '12 at 13:24
Thank you for your answer. Uniformly equivalent it shall be. –  Daniel Mansfield Jul 25 '12 at 1:07

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