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, I 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?

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?

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, I define the function $F:[0,1]\to[0,\infty)$ by $$ F_{\mu/\nu}(x) = \inf\{\nu(A): A\in\mathscr{B},\mu(A)\le x \}.$$

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

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, I 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?