A personal annotations of Veach's thesis (15) p.168

p.168 D3 and Equation 5.30 updated

I got a comment about Equation 5.30 from my friend Daniel. However, Equation (1) has absolute value operator at |f'(x_0)|, I don't understand this yet. To see this Equation, we could think about this is a chain rule. If we write this in an integral form (as the substitution rule) as in the Equation (2). This is substantially equivalent with the chain rule. But still, the domain doesn't match with Equation 5.30.

On the other hand, if we read the thesis until p.170, Equation 5.35 is a definition and this looks like showing a linearity. It is Equation (3). Maybe the question is, what is the linear coefficient a_{\beta}. That's my guess. What coefficient makes this equation consistent through the Dirac's \delta, that could be the definition of Equation 5.35.

This has an advantage, if this can be, we don't need integrate every time, this \delta looks like just an ordinary function. This is convenient. (as the Veach said in the text, this kind of stuff is the great part of this paper.)

The following is a slightly changed from the paper, but, basically the same with p.170's Equations. Notice, \beta is a bijective function and \beta(x) = x', x = \beta^{-1}(x'),
Therefore, comparing the first equation and the last one, we have
Yes, finally we got it in this way. In the paper, this x and x' replacement is not at one point, maybe some reason, but, for my understanding, I replace them at once.

In the end, the domain of measure linearly changed. This is interesting. But, integration operator is a linear operator, maybe it is natural. Still, this result is interesting.

The last question, why |f'(x_0)| has the absolute operator? This Equation 5.30's  question still remains for me. But it is much better, I think I almost see it. (or I see an illusion as usual...)

I thank Daniel S. to give me hints of this question. I also thank his great patience on my stupid basic questions.

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