(Novel?) notion of concentration/dispersion 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?
 A: 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.
