1
vote
1answer
44 views

The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
1
vote
1answer
173 views

A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
4
votes
1answer
349 views

Multiplicative Haar measure on $\mathbb{Q}_p$?

I have read in a book that if one takes $\mu$ to be the additive Haar measure on $\mathbb{Q}_p$, the p-adic rationals, then $$\nu(A) := \int_{A} 1/|x|_p dx$$ is a multiplicative Haar measure on ...
7
votes
2answers
314 views

What does the Haar measure on $\hat{\mathbf{Z}}$ look like?

What does the Haar measure on $\hat{\mathbf{Z}} = \prod_p \mathbf{Z}_p$ look like? Does it bear any relation to the "upper density" of a subset $S\subset\mathbf{N}$ defined by $m(S) = ...
2
votes
0answers
142 views

Can this function extend to a measure?

I believe I have one final question on a function I've been thinking about. To set up, let $\mathbb{Q}_p$ be the $p$-adic numbers, and let $B(x,r)$ denote the closed balls $$ ...
4
votes
2answers
199 views

Can I build a finitely additive function on $\mathbb{Q}_p$?

This is partially motivated by a question I saw earlier here, Does such a finitely additive function exist? I've been reading about the topology of $\mathbb{Q}_p$ in Knapp's Advanced Algebra in ...