15
$\begingroup$

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:

  1. Do these ultrametric integrals have real analogues like: being a limit of Riemannian sums or an operation inverse to differentiation?
  2. Are they compatible to each other? Is any of them a generalization of the others?
  3. What are the positive and negative attributes of the cited definitions?
  4. Where can I find an exhaustive table of integrals? I'm mainly interested in something similar to https://www.tug.org/texshowcase/cheat.pdf.
  5. Can we somehow imitate the Lebesgue integration theory?
$\endgroup$
  • 2
    $\begingroup$ For (5) see Schikhof "Ultrametric Calculus - An Introduction to p-adic Analysis" where it is shown that the approach to integration based on $\sigma$-algebras doesn't generalize to the $p$-adic setting - maps $\mathbb Z_p\to\mathbb Q_p$ - but the Riesz representation approach does generalize well. He points to van Rooij's "Non-Archimedean Functional Analysis" for further theory. $\endgroup$ – Dap Mar 28 '18 at 18:07
8
+75
$\begingroup$

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...

$\endgroup$
  • 1
    $\begingroup$ The Volkenborn integral is not translation invariant: the Wikipedia article mentions this, or consider the difference between the integrals of $x$ and $x+1.$ Woodcock's "An Invariant $p$‐adic Integral on $Z_p$" claims it can be understood as an invariant integral if the values are taken in $C_{p^\infty}$ (the same as $\mathbb Q_p/\mathbb Z_p$?), but I don't understand that $\endgroup$ – Dap Mar 28 '18 at 15:46
  • 1
    $\begingroup$ Another point worth mentioning. The Volkenborn integral isn't a measure i.e. bounded on compact sets, since the indicator function of the subset $p^n\mathbb Z_p\subset \mathbb Z_p$ integrates to $p^{-n}$ $\endgroup$ – Dap Mar 28 '18 at 15:59
  • $\begingroup$ @Dap Concerning invariance, you are right, I will correct this. But isn't it invariant by translations when $f$ is locally constant ? $\endgroup$ – user120527 Mar 28 '18 at 16:02
  • $\begingroup$ yes I see what you mean, that's true $\endgroup$ – Dap Mar 28 '18 at 16:03
  • $\begingroup$ That clarifies more than just a bit (+1), although of course more can be desired. (Tiny correction, in your second formula under the second point, the "$n$" under the sum and in the function should probably be an $i$ or something like that.) $\endgroup$ – Torsten Schoeneberg Apr 1 '18 at 5:02
3
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.