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

The link to the page.

I do not understand the following.

  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?

I tried understanding this using examples, but I still could not form a clear understanding. Is the proof incomplete as given on this page, or am I missing some commonly assumed facts?

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 $A$?

Update

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.

enter image description here

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