# Standard notation or name for the measure defined by an integral over a function

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.

• I think it was induced measure, but I am not sure.
– Jan
Jan 13 '17 at 22:52
• Universal notation: $$d\nu=f\cdot d\mu$$ (One can be surprised by the suggestions made in the answers below.)
– Did
Mar 26 '17 at 21:19
• Ok, thank you for this comment. Mar 26 '17 at 21:22

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.

• 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. Jan 13 '17 at 23:34

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.

• I don't like $f \odot \mu$, since this reminds me more of a product than an integral. Jan 13 '17 at 23:46
• 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. Jan 13 '17 at 23:47
• 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. Jan 13 '17 at 23:59