Understanding the proof of Basu's theorem

In understanding the proof of Basu's thereorem, from wikipedia, I could not understand a few steps. Here's the wiki page.

1. We are not making any assumptions on the probability density function. Hence, what does $$P_\theta$$ and $$P$$ represent individually? This also seems to explain why $$P_\theta (...|T=t)$$ does not depend on $$\theta$$, doesn't it?
2. How is $$P_\theta^A(B)=P_\theta (A^{-1}B)$$? Further, what does the latter quantity indicate?
Furthermore, just to clarify once, if $$A$$ is ancillary to $$T$$, the latter being a sufficient statistic for parameter $$\theta$$, all other parameters necessary to characterise the probability function must be sufficiently predicted by $$A$$?
The wikipedia page got updated after that, with some additional changes. But I still fail to understand what all these quantities represent (especially $$P$$, $$P_\theta$$,$${P_\theta}^A$$). Also, if someone can explain the procedure in more explicit terms and relatively expanded explanations.