What is the compelling need for introducing a theory of $p$-adic integration?
Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a Lebesgue measure on $p$-adic spaces, and just integrate real or complex valued functions on $p$-adic spaces, or is something more possible like integrating $p$-adic valued functions on $p$-adic spaces? What is the machinery used?
Then again, does the integration on spaces like $\mathbb C_p$ give something more than the usual integration in real analysis? I mean, the integration of complex valued functions of complex variables, or more precisely holomorphic functions, is much a much more interesting topic than measure theory. Is a similar analogue true in $p$-adic cases?
I have also seen mentioned that Grothendieck's cohomology theories like etale cohomology, crystalline cohomology etc., fit into such $p$-adic integration theories. What could possibly be the connection?