I have questions about the proof of the following Proposition from the book Introduction to the Theory of Non-Symmetric Dirichlet forms.

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First, how do we get the independence of $G_\alpha (B)$ of $\alpha$ by 1.4(iii) and that each $G_\alpha$ is one-to-one by 1.4(i) and 1.4(iii)?

Second, once we conclude that $\alpha - G_\alpha^{-1} = \beta - G_\beta^{-1}$, how do we get that $D(L)=G_\alpha (B)$ is dense in $B$ by 1.4 (i)?

I would greatly appreciate any help.


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