TITLE: Various $p$-adic integrals
QUESTION [17 upvotes]: Here are three (possibly different) definitions of $p$-adic integrals that I have encountered during my self-studies.
First of all, here is what Vladimirov, Volovich and Zelenov write at the beginning of their book on mathematical physics:

As the field $\mathbb Q_p$ is a locally compact commutative group with
  respect to addition then in $\mathbb Q_p$ there exists the Haar
  measure, a positive measure $dx$ which is invariant to shifts. We
  normalize the measure $dx$ such that $\int_{|x|_p \le 1} dx = 1.$
  Under such agreement the measure is unique.
For any compact $K \subseteq \mathbb Q_p$ the measure $dx$ defines a positive linear
  continuous functional on $C(K)$ by the formula $\int_K f(x) dx$. A
  function $f \in \mathcal L^1_{\textrm{loc}}$ is called integrable on
  $\mathbb Q_p$ if there exists $$\lim_{N \to \infty} \int_{B(N)} f(x)dx = \lim_{N \to \infty} \sum_{\gamma = -N}^\infty \int_{S(-\gamma)} f(x) dx.$$

Their denote by $B(N)$ the set $\{x \in \mathbb Q_p : |x|_p \le p^N \}$ and $S(\gamma)$ is defined as $B(\gamma) \setminus B(\gamma-1)$.
Next we have the famous Volkenborn integral, described as follows by Robert:

We say that $f$ is strictly differentiable at a point $a \in X$ - and
  denote this property by $f \in \mathcal S^1(a)$ - if the difference
  quotients $[f(x) - f(y)]/(x-y)$ have a limit as $(x, y) \to (a,a)$
  ($x$ and $y$ remaining distinct). By the way, $\mathcal S^1(X) := \bigcap_{a \in X} \mathcal S^1(a)$.
  The Volkenborn integral of a function $f \in \mathcal S^1(\mathbb Z_p)$ is by definition $$\int_{\mathbb Z_p} f(x) dx = \lim_{n \to
 \infty} \frac{1}{p^n} \sum_{j=0}^{p^n-1} f(j).$$

Finally a quote from Koblitz book "$p$-adic numbers, analysis and $\zeta$-functions":

Now let $X$ be a compact-open subset of $\mathbb Q_p$. A $p$-adic distribution $\mu$ on $X$ is a $\mathbb Q_p$-linear vector space homomorphism from the $\mathbb Q_p$-vector space of locally constant functions on $X$ to $\mathbb Q_p$. 

Later he states that for $p$-adic measures (distributions that are bounded on compact-open subsets by some constant) and continuous functions $f$ there is a reasonable way to define $\mu(f) =: \int_X f \mu$.
Cassels is confusing me even more as he mentions Shnirelman. So, here are my actual questions:

Do these ultrametric integrals have real analogues like: being a limit of Riemannian sums or an operation inverse to differentiation?
Are they compatible to each other? Is any of them a generalization of the others?
What are the positive and negative attributes of the cited definitions?
Where can I find an exhaustive table of integrals? I'm mainly interested in something similar to https://www.tug.org/texshowcase/cheat.pdf.
Can we somehow imitate the Lebesgue integration theory?

REPLY [6 votes]: 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".