Consider the measurable space $(\Omega,\mathscr{B})$ endowed with two positive measures: a "volume $\nu$" and a probability measure $\mu$. For example, one might take $\Omega=\mathbb{R}^n$ (with the usual $\sigma$-algebra) and $\nu$ as the Lebesgue measure.


I would like to quantify the notion of "a high proportion of $\mu$'s mass is concentrated on a region of small volume". To that end, let us define the function $F:[0,1]\to[0,\infty)$
by
$$ F\_{\mu/\nu}(x) = \inf\{\nu(A): A\in\mathscr{B},\mu(A)\ge x
\}.$$


Question: Is this (or perhaps a closely related) notion known?



(Novel?) notion of concentration/dispersion
The function you propose is related to the L'evy concentration function, 
 studied by Kolmogorov, Rogozin, Esseen
and others. See the special volume [1] <https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70>


The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].


Returning to the original question, Decompose $\mu=\mu\_a+\mu\_s$ where $\mu\_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu\_s$ is singular to $\nu$. Write $M\_s$ for the total mass of $\mu\_s$. 
If $M\_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A\_c:=\{f>c\}$.
If there is such a set with $\mu(A\_c)=x-M\_s$ then $F(x)=\nu(A\_c)$. If there is no such $c$, find the infimmum $c\_\*$ of the constants $c$ such that $\mu(A\_c)<x-M\_s$. one needs to do some tie-breaking inside the level set where $f=c\_\*$ and take a subset there of suitable $\mu$ measure $x-M\_s-\mu(A\_{c\_\*})$.
Thus $F(x)=\nu(A\_{c\_\*})+(x-\nu(A\_{c\_\*}))/c\_\*$.


[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.


[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.


[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.


[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.


[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.


