The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta\_n(x)$ spiked at the the origin, in the sense that:


$\lim\_{n \rightarrow \infty} \int\_{\mathbb{R}} f(x)\delta\_n(x-a)dx = f(a)$


I am aware that one can define a $p$-adic Dirac measure satisfying the sifting property


$\int\_{\mathbb{Z}\_p} f(x) d\delta\_p = f(0)$


for functions $f:\mathbb{Z}\_p \rightarrow \mathbb{R}$ (or $\mathbb{C}$). 


**QUESTION**: Can the $p$-adic Dirac measure also be seen as a weak limit of $p$-adic functions as in the real case? That is, do there exist functions $\delta\_n(x):\mathbb{Z}\_p \rightarrow \mathbb{R}$ such that


$\lim\_{n \rightarrow \infty}\int\_{\mathbb{Z}\_p} f(x)\delta\_n(x-a) d\mu = f(a)$


for any suitably well behaved function $f(x)$, where $d\mu$ is the Haar measure? Or if that is not the appropriate analogue, what would be?


Apologies if this is standard, but I couldn't find a reference for this in the literature after some searching. Thanks.



$p$-adic Dirac measure as a weak limit
The answer is yes. Moreover:


1) For every (first countable) locally compact group the $\delta$ measure at the identity is a weak limit of (a sequence of) continuous functions. You can find this in any abstarct harmonic analysis text, eg Folland, "A course in abstract harmonic analysis" Prop 2.42. Look up: *approximate identity*.


2) Explicitly for $\mathbb{Z}\_p$ you can take the functions $f\_n=p^n\chi\_{p^n\mathbb{Z}\_p}$.


3) This specific construction could be generalized to any totally disconnected locally compact group: each such group has a basis of identity nbd consisting of compact open subgroups - take the normalized charcteristic functions of these. For the general case look up:*van Danzig Theorem*. For compact groups look up: *profinite topology*.


