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In the ye olden days of intro calculus, we are taught that some integrals which are difficult, like

$$ \iint_{\|(x,y)\| \leq 1} \sqrt{x^2+y^2} ~dx~dy $$

can be turned into something nicer by using the substitution $x = r\cos(\theta)$, $y=r\sin(\theta)$. Then this integral turns into

$$ \int_{0}^{2\pi}\int_{0}^{r} r^2~dr~d\theta $$

because the Jacobian when switching coordinate systems is $dxdy = rdrd\theta$. But eventually, one comes across the usage of arbitrary measures (and in particular, the Lebesgue measure) as a meaningful tool of integration. In situations where the Riemann integral existed, everything's the same, but now you can integrate even more, woohoo!

But what happens now? Is it always the case that when converting between coordinate systems with some measure

$$ \iint f(x,y)~d\mu d\nu$$

there exists some sensible Jacobian between our measures? Are there cases where two measures are completely incompatible, and there is no way to convert between them when doing a coordinate transformation?

IdenticallyEulerian
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  • See this page. Note in particular that changing variables for a double Lebesgue integral as you have above is equivalent to changing variables for the single Lebesgue integral with respect to the product measure $\mu\otimes\nu$. – csch2 Mar 24 '23 at 18:17

1 Answers1

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Is it always the case that when converting between coordinate systems with some measure there exists some sensible Jacobian between our measures?

For the Jacobian to exist, the first order partial derivatives must exist. In your example (switching between Cartesian and polar coordinates), the first order partial derivatives clearly exist because there is a differentiable tranformation $(x,y) \rightarrow (r cos(\theta), r sin(\theta))$. Depending on your application/problem, you might also like this to be an invertible map, so you can convert back to the original coordinate system. For your example, we have $r = \sqrt{x^2 + y^2}$ and $\theta = tan^{-1}(\frac{y}{x})$. These are also differentiable.

Are there cases where two measures are completely incompatible, and there is no way to convert between them when doing a coordinate transformation?

I'm sure there are some pathological measures that you haven't encountered yet. I recommend checking out the Dirac measure and the discrete measure. I'm not sure what it would look like to "convert" between measures like this, but I think you might be able to find an answer by looking into these. Good luck!

jtb
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