Various $p$-adic integrals

Not really an full answer, but some comments (that hopefully answer some of your queries).

There seems to be a big confusion here : what do we want to integrate, i.e. to define $\int_{\mathbb{Z}_p} f(x)dx$, what is the 'nature' of $f$, and of the result ?

• For the first one, (it's the one I am familiar with), the Haar measure on $\mathbb{Z}_p$ is in particular a map $\mu$ that assign to a Borel subset (say $E$) of $\mathbb{Z}_p$ real number $\mu(E)$. Take $f=1_E$ the characteristic function of $E$, then $$\int_{\mathbb{Z}_p} 1_E d\mu=\mu(E) \in \mathbb{R},$$ by definition. The result is then a real number. That tells us that measure theory is about integration of functions with value in $\mathbb{R}$ !

  That means that in this context, for example, $\int_{\mathbb{Z}_p} xd\mu(x)$ has no meaning from the point of view of this definition of integral.

  If one is to look for an analogue of Riemann sums, we may notice that the sequence $(n)_{n\in \mathbb{N}}$ is equidistributed on $\mathbb{Z}_p$ with respect to $\mu$ (An ergodic theorist would say that the transformation $T:\mathbb{Z}_p\to \mathbb{Z}_p, T(x)=x+1$ is uniquely ergodic). The analogue of Weyl's criterion holds: for $f$ a real-valued continuous function, bounded on $\mathbb{Z}_p$, then $$\int_{\mathbb{Z}_p} fd\mu=\lim_{N\to +\infty} \frac1N \sum_{n=0}^{N-1} f(n).$$

  Another thing: the formula for integration on $\mathbb{Q}_p$ in the OP is in this case an analogue of the definition of the improper integral: $$\int_{-\infty}^{\infty} fdx := \lim_{T\to +\infty} \int_{-T}^T f(x)dx,$$ which allows to make sense of functions whose integral in the measure theory sense has no meaning, like $f(x)=\sin(x)/x$.

• About the Volkenborn integral, although this is not said, if we are to believe https://en.wikipedia.org/wiki/Volkenborn_integral, is made to integrate functions $f$ with values in $\mathbb{C}_p$ (but let's reduce it to functions with values in $\mathbb{Q}_p$ to be able to compare with the latter definition). The definition given in the OP $$\int_{\mathbb{Z}_p} f(x)dx:=\lim_{N\to +\infty} \frac1{p^N} \sum_{n=0}^{p^N-1} f(n),$$ looks pretty much the same than the analogue of Riemann sums above, but with one big difference: one looks only at the subsequence $(p^N)_{N\geq 0}$. Indeed, an easy but instructive example is to compute these Riemann sums for $f(x)=x$. Then $$\frac1{n} \sum_{k=0}^{n-1} f(k)=\frac{n-1}2 \in \mathbb{Q}_p,$$ which does not converge (recall the topology is the $p$-adic one), but does for the subsequence $(p^k)_{k\geq 0}$, to $-1/2$. This gives us $\int_{\mathbb{Z}_p} xdx=-1/2$, as said in the above wikipedia link (which contains, btw, a few formulas as required).

• The third definition deals with $\mathbb{Q}_p$-linear vector space homomorphism of locally constant functions to $\mathbb{Q}_p$. So clearly here we are also looking at integration of function $f:\mathbb{Z}_p\to \mathbb{Q}_p$. The Volkenborn integral is an example, locally constant functions being strictly differentiable. But it's not the only one (EDIT : I previously stated that the Volkenborn integral is invariant by translations, which was wrong, as noted by Dap). So this definition is more general (the dirac measures works, for example).

  I hope this clarifies a little bit...

This is just a comment after @user120527's answer. The Volkenborn integral and $p$-adic measures are special cases of "$p$-adic distributions", which are defined as elements of topological dual spaces of nice functions. Let $K$ be a closed subfield of $\mathbb{C}_p$ (the completion of the algebraic closure of $\mathbb{Q}_p$), and let $$C^0(\mathbb{Z}_p,K)=\{f:\mathbb{Z}_p\to K \text{ continuous}\},$$ $$C^1(\mathbb{Z}_p,K)=\{f:\mathbb{Z}_p\to K \text{ strictly differentiable}\}.$$ Then, the Volkenborn integral $f\mapsto\int_{\mathbb{Z}_p}f(t)dt$ is an element of the topological dual of the space $C^1(\mathbb{Z}_p,K)$. Also, a $p$-adic measure $\mu$ is just an element of the topological dual of $C^0(\mathbb{Z}_p,K)$, by means of $$\mu:f\mapsto \mu(f)=\int_{\mathbb{Z}_p}f(t)\mu(t).$$

In general, a $p$-adic distribution is an element of the dual of the space of locally analytic functions $\mathbb{Z}_p\to K$, which can be extended to a nicer space, such as $$C^r(\mathbb{Z}_p,K)=\{f:\mathbb{Z}_p\to K \text{ $r$-th times strictly differentiable}\}.$$

This "$p$-adic dual theory" was developed by Yvette Amice. For a very nice article (with full proofs) of this theory, see "Fonctions d'une variable p-adique" by Pierre Colmez: http://webusers.imj-prg.fr/~pierre.colmez/fonctionsdunevariable.pdf

Finally, the Shnirelman integral is not a $p$-adic distribution. One may think of $p$-adic measures as analogues of the Riemann integral, and the Shnirelman integral as an analogue of the complex line integral. Neal Koblitz treats the Shnirelman integral in his book "P-adic Analysis: A Short Course on Recent Work".

This is a beautiful theory with many arithmetical applications. Good luck studying it!

PS: I don't know too much about complex valued $p$-adic integration, but for the Haar measure case over local fields, the keywords are "Tate's thesis".