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?
Best Answer
I would normally take $p$-adic integration to mean "integration of $p$-adic valued functions" or "integration of differential forms with some kind of $p$-adic valued functions as coefficients", where the integration is also taking place over some kind of $p$-adic space or manifold.
The reason for wanting such theories are various. One reason is indicated in George S.'s answer: there are known analogues of classical Hodge theory, known as $p$-adic Hodge theory, whose proofs however are not analytic, but rather proceed via arithmetic geometry. One would like to have more analytic ways of thinking about them, and this is one goal of Robert Coleman's theory. (In a recent volume of Asterisque, namely vol. 331, Coleman and Iovita have an article, "Hidden structures on semistable curves", related to this problem.) (Note also that $p$-adic Hodge theory relates $p$-adic etale cohomology to crystalline cohomology, which gives on answer to your question of how $p$-adic integration might be related to those topics.)
Another reason is that many integral formulas (involving usual archimedean integrals) appear in the theory of classical $L$-functions attached to automorphic forms, and one would like, at least in certain contexts, to be able to write down $p$-adic analogues so as to construct $p$-adic $L$-functions.
As for what machinery is used: in the theory of $p$-adic $L$-functions and related contexts in Iwasawa theory, often nothing more is used than basic computations with Riemann sums. In the material related to $p$-adic Hodge theory, much more substantial theoretical foundations are used: tools from arithemtic geometry, rigid analysis, possibly Berkovich spaces, and related topics.