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Why does the equation at the bottom of the following page hold?

http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/Transforming_between_Distributions.html

$p(\theta,\phi)d\theta d\phi=p(\omega)d\omega$

Is this a conclusion of probability and measure theory?

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  • $\begingroup$ Right above it you have $d\omega = \sin\theta \,d\theta\,d\phi$. Substitute and see what you get. $\endgroup$
    – lightxbulb
    May 25 '20 at 13:20
  • $\begingroup$ I think the story should be like this: Premises: $d\omega=sin\theta\ d\theta\ d\phi$ and $p(\theta,\phi)d\theta\ d\phi=p(\omega)d\omega$; Conclusion: $p(\theta, \phi)=sin\theta\ p(\omega)$. $\endgroup$
    – chaosink
    May 25 '20 at 16:33
  • $\begingroup$ That's it really. And yes, the above is from probability and measure theory. You can define your pdf with respect to some measure and then find its expression with respect to a different measure. $\endgroup$
    – lightxbulb
    May 25 '20 at 17:04
  • $\begingroup$ But I couldn't find a formal statement in math books. Could you give a reference? Thanks! $\endgroup$
    – chaosink
    May 25 '20 at 17:15
  • $\begingroup$ You need a statement about what? That $d\omega = \sin\theta \,d\theta\,d\phi$? That's by definition of solid angle. But if you want to get where the sine comes from, you could just compute the coordinate transformation from cartesian to spherical coordinates and then set $r=1$ ($r^2\sin\theta$ pops from the Jacobian of the transformation). $\endgroup$
    – lightxbulb
    May 25 '20 at 20:19

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