The statistical average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant

Question

I am trying to solve a Lagrange multiplier problem for the following equation

$$ L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \right) +\beta \left( \overline{O} - \int_{-\infty}^\infty O(x) \rho(x) dx \right) \tag{1} $$

for $\frac{\partial L}{\partial \rho(w)} =0$.

Where the first term is the relative entropy, the second term is the constraint that it sums to one, and the last term is the constraint of an observable involving its average value. The problem solves for the probability measure that maximizes the relative entropy of a continuous parametrization x.

Eventually, I wish to apply the probability measure to a curved manifold. This is why coordinate invariance is very interesting to me.

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An interesting property of the relative entropy is that its equations remains invariant with respect to a change in variable. Indeed,

$$ \int_{-\infty}^\infty \rho(x)\ln \frac{\rho(x)}{q(x)} dx \to \int_{-\infty}^\infty \rho(y(x)) \left|\frac{\partial y}{\partial x}\right|\ln \frac{\rho(y(x))\left|\frac{\partial y}{\partial x}\right|}{q(y(x)) \left|\frac{\partial y}{\partial x}\right|} dx = \int_{-\infty}^\infty \rho(y)\ln \frac{\rho(y)}{q(y)} dy \tag{2} $$

and

$$ \int_{-\infty}^\infty \rho(x)dx \to \int_{-\infty}^\infty \rho(y(x)) \left|\frac{\partial y}{\partial x}\right| dx = \int_{-\infty}^\infty \rho(y) dy \tag{3} $$

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However, the last term isn't. Indeed:

$$ \int_{-\infty}^\infty O(x) \rho (x) dx \to \int_{-\infty}^\infty O(y(x)) \left|\frac{\partial y}{\partial x}\right| \rho (y(x)) \left|\frac{\partial y}{\partial x}\right| dx = \int_{-\infty}^\infty O(y) \rho (y(x)) \left|\frac{\partial y}{\partial x}\right| dy \tag{4} $$

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How can I modify the integral that contains O so that it is coordinate invariant?

What is the expression for the average of a function O(x) over a continuous parametrization x, such that the average value is coordinate invariant?

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edit:

It appears to me that the solution is to consider the observable to be the ratio between two quantities:

$$ \overline{O} = \int_{-\infty}^\infty \frac{A(x)}{B(x)} \rho(x) dx $$

where $O(x) := A(x)/B(x)$. In this case, it is invariant for the same reason that equation (2) is:

$$ \int_{-\infty}^\infty \frac{A(x)}{B(x)} \rho(x) dx \to \int_{-\infty}^\infty \frac{A(y(x)) \left| \frac{dy}{dx} \right| }{B(y(x)) \left| \frac{dy}{dx} \right|} \rho(y(x)) \left| \frac{dy}{dx} \right| dx = \int_{-\infty}^\infty \frac{A(y) }{B(y) } \rho(y) dy $$

Can anyone weight in?