Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1] \cup {\rm Im} \, [f_2],$ the distributions are denoted $\rho_{1,2}$, respectively.

My questions:

1. Is there a sense of distance between $\rho _1$ and $\rho _2$ that does not involve their explicit computation, but only of the functions $f$?
2. Does that answer changes if both $f$ are smooth?
3. If both their images are bounded?
4. If $f_1(x) = x$, and so $\rho _1$ is the uniform distribution?

My motivation: Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

Disclaimer: Having a "good distance measure" is vague, and can be interpreted in many ways. I know. But the ways to define it I already know involve the computation of $\rho$, so I'd rather leave it open-ended for now.

Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1] \cup {\rm Im} \, [f_2],$ the distributions are denoted $\rho_{1,2}$, respectively.

My questions:

1. Is there a sense of distance between $\rho _1$ and $\rho _2$ that does not involve their explicit computation, but only of the functions $f$?
2. Does that answer changes if both $f$ are smooth?
3. If both their images are bounded?
4. If $f_1(x) = x$, and $d\mu = dx$, and so $\rho _1$ is the uniform distribution?

My motivation: Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

Disclaimer: Having a "good distance measure" is vague, and can be interpreted in many ways. I know. But the ways to define it I already know involve the computation of $\rho$, so I'd rather leave it open-ended for now.