The collision rate probability decreases when using compression? Some standard compression procedures, like the  IANA's gzip 
provided by HTTP protocol, will consume CPU-time anyway...
So we can reuse compressed file, $Z(x)$, in the checksum procedure. 
That is, we can use $H(Z(x))$ instead $H(x)$.

Let $H$ be a collision resistant hash function and $P_c[H](S)$ 
the collision rate probability
about a sample set $S$ of input elements (eg. random numbers). 
It decreases when using $Z(x)$? $P_c[H\circ Z](S) \le  P_c[H](S)$ &nbsp; ?



Intuitivally we can imagine some decrease in collision-rate because hashing a file is similar to an aleatory sampling procedure: 
if we reduce the redundance in the sample set, we reduce collision probability.
HTML, XML, and many other file formats have a lot of redundancy.

If the intuition is correct, the questions is also about "how mutch better is it"? 



NOTICE: I am using the strange term collision rate probability (with a link to this other question) to avoid confusion with the "general collision probability". Please take care to not mix the concepts in the answer.
No. On the contrary, if we model $H$ as a random function chosen independently of $Z$ and $S$, then $P_c[H\circ Z](S)=P_c[H](S)$.

Argument is that $P_c[H](S)$ is dependent only on the width of $H$ and the number of elements in $S$, as long as $H$ is modeled as a random function chosen independently of $S$. Define $S'$ as the set of $Z(s)$ for $s$ in $S$. It holds that $P_c[H\circ Z](S)=P_c[H](S')$, and because compression is reversible, $S'$ and $S$ have the same number of elements.