Equip the additive group $\mathbb{R}/\mathbb{Z}$ with the discrete topology. What is its Pontryagin Dual?

Question

Just like the title says, what is the Pontryagin Dual of $\mathbb{R}/\mathbb{Z}$, when we equip it with the discrete topology? In particular, since this dual must be compact, what is the formula for its Haar probability measure (in terms of tensor products of p-adic measures or what-have-you), and, what is the formula for the duality pairing—that is, the bilinear map $f:\mathbb{R}/\mathbb{Z}\times\widehat{\mathbb{R}/\mathbb{Z}}\rightarrow\mathbb{R}/\mathbb{Z}$, so that every character on $\mathbb{R}/\mathbb{Z}$ is of the form $x\mapsto e^{2\pi if\left(x,\xi\right)}$ for some $\xi\in\widehat{\mathbb{R}/\mathbb{Z}}$?

Answer 1

As a discrete group, $G=\mathbb R/\mathbb Z$ is isomorphic to $\mathbb Q/\mathbb Z\times H$ where $H$ is a huge coproduct of copies of $\mathbb Q$. The dual of the first factor is $\widehat{\mathbb Z}$, the product of all the various $p$-adic integers (for all $p$) $\mathbb Z_p$. This has a familiar Haar measure, etc. Since $\mathbb Q$ is an ascending union (colimit) of ${1\over N}\mathbb Z$'s, its dual is the limit of the duals of those limitands, namely, a limit of larger-and-larger circles, so is $\mathbb A/\mathbb Q$, where $\mathbb A$ is the rational adeles. Each of these has its usual (quotient) Haar measure.