I started reading the [Lectures on Condensed Mathematics][1]. I am looking at the material at page 32-34. I have three fundamental computation questions: 

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 1. At the last line of pg 32 - it seems to imply that for finite sets $S$,  $\Bbb Z[S] \simeq \underline{Hom}(C(S, \Bbb Z ), \Bbb Z) $? 

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2. In pg 33 line 4 how is $Hom(C(S,\Bbb Z), \Bbb Z)$ identified with measure on $S$? 
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 3. In proof of Proposition 5.7, there was the following equivalence, 

$$ RHom(\prod_J \Bbb R, \Bbb Z) \simeq RHom_{\Bbb R} (\prod_J  \Bbb R, R\underline{Hom}(\Bbb R, \Bbb Z) )=0$$

why is this true and what exactly does $RHom_{\Bbb R}$ mean? Is there some adjunction happening here from $\Bbb Z$ modules of $Cond(Set)$ to $\Bbb R$-modules of $Cond(Set)$?  

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I would appreciate if there are some related references for the general set up in 2 and 3.  

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For 1, I would like to compute the $T$ points for $T$ extremely. disconnected. 
What i don't see is an easy expression for lhs.  
$$\Bbb Z[S] (T)= \bigoplus_{C(T,S)}\Bbb Z$$
 Conversely for rhs we have 

$$\underline{Hom}(C(S,\Bbb Z), \Bbb Z)(T)=Hom(\Bbb Z[T] \otimes C(S,\Bbb Z), \Bbb Z) $$
This doesn't seem an easy expression to handle too. 


  [1]: https://www.math.uni-bonn.de/people/scholze/Condensed.pdf