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Let $(\Omega,\mathcal{F},\mu)$ be a measure space. For a measurable function $f \geq 0$ one can define a measure $\nu$ by $$ \nu(M) : = \int_M f d \mu = \int_{\Omega} f \mathbb{1}_{M} d \mu $$ for $M \in \mathcal{F}$. Is there a standard notation/name for the measure $\nu$? I think I saw something like $f \odot \mu$ but I am not sure.

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  • $\begingroup$ I think it was induced measure, but I am not sure. $\endgroup$
    – Jan
    Jan 13 '17 at 22:52
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    $\begingroup$ Universal notation: $$d\nu=f\cdot d\mu$$ (One can be surprised by the suggestions made in the answers below.) $\endgroup$
    – Did
    Mar 26 '17 at 21:19
  • $\begingroup$ Ok, thank you for this comment. $\endgroup$
    – user148692
    Mar 26 '17 at 21:22
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The name given to a map from a function to some field (usually the reals) obtained by using that function in some sort of operator is a functional (or sometimes a functor).

The notation often uses square brackets, as in $\nu_M[f]$. The branch of math this often appears in is the calculus of variations.

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  • $\begingroup$ I came across with this in the Radon–Nikodym theorem. So you suggest writing $\mu_M[f] = \int_M f d\mu$? Do you have a reference for this? I would prefer a measure-theory or probability-related book. $\endgroup$
    – user148692
    Jan 13 '17 at 23:34
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I have seen it used in the form $f\odot \mu$ in probability theory, of course for the measure that is absolutely continuous with respect to $\mu$ with Radon-Nikodym derivative $f$. But it is not a very nice notation, since it misses the right form $d\mu$.

It is common (without using your notation!) in calculus of variations (Young measures), ergodic theory (equilibrium measures), multifractal analysis (Gibbs measures), delay equations (bounded variation functions and Riemann-Stieltjes), etc. But I recall no notation being used that could be widely accepted.

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  • $\begingroup$ I don't like $f \odot \mu$, since this reminds me more of a product than an integral. $\endgroup$
    – user148692
    Jan 13 '17 at 23:46
  • $\begingroup$ I also don't. I would suggest simply $\mu_f$. Perhaps it is not common but it is the simplest possible! PS: I have seen it written (in Bowen's little book) as $f\mu$. Not nice either. $\endgroup$
    – John B
    Jan 13 '17 at 23:47
  • $\begingroup$ I like your suggestion $\mu_f$ most since it is short, simple and just indicates $\mu_f$ is a measure depending on $\mu$ and $f$. I think also $\mu(f) = \int_{X} f d\mu$ is quite common and with your suggested notation we would have $\mu(f) = \mu_f(X)$. Now that you mention it, I think I saw $f\mu$ to notate e.g. densities with respect to a measure. But again the notation is not the best. $\endgroup$
    – user148692
    Jan 13 '17 at 23:59

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