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?