MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.