This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic.

Since $\mathbb{Q}_p$ is a locally compact group with $\mathbb{Z}_p$ as a compact subset, there exists a unique Haar measure $\mu$ on $\mathbb{Q}_p$ with $\mu(\mathbb{Z}_p)=1$.

The Volkenborn integral for functions $\mathbb{Z}_p\to\mathbb{Q}_p$ is defined by $$ \int_{\mathbb{Z}_p} f = \lim_{n\to\infty}\frac{1}{p^n}\sum_{x=0}^{p^n-1}f(x), $$ takes values in $\mathbb{Q}_p$, and is not translation invariant.

But now suppose I define an integral over $\mathbb{Z}_p$ for functions $\mathbb{Z}_p\to\mathbb{R}$ using the same expression as the Volkenborn integral. Then it seems that I can integrate the functions $\mathbb{1}_{a+p^n\mathbb{Z}_p}$ to get $\frac{1}{p^n}=\mu(a+p^n\mathbb{Z}_p)$, and because the integral so defined is clearly ($\mathbb{R}$-)linear in $f$, it would appear that it agrees with the integral over the Haar measure (at least when both exist).

Since such a Volkenborn-style integral for $\mathbb{R}$-valued functions is never discussed in the literature that I could find, I wonder if I am missing some obvious problem with the construction here, or if this is just a well-known rarely-mentioned fact.

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