I've solved this exercise somewhat.To complete it please help me
Haar measure on G $=$ translation invariant on G
$$ μ(A)=μ(A+t)$$ if $ G=\mathbb{R}$ then Haar measure on G is lebesgue measure. and if
$G=\mathbb{T}$ then Haar measure on G is $dt/2π$. I know that $μ_\mathbb{R}× μ_\mathbb{T}$ is a
Haar measure on product group $\mathbb{R} × \mathbb{T}$ .
$$ f : \mathbb{R} × \mathbb{T}→\mathbb{C}\quad by \quadμ(f)= \int_{\mathbb{R}}\int_{\mathbb{T}} f(x,t) (dt/2π) ~ \mathrm{d}{x}. $$
Let $G=\mathbb{R} × \mathbb{T}\quad,\widehat{\mathbb{G} }=\widehat{\mathbb{R} × \mathbb{T}}=\widehat{\mathbb{R} }×\widehat{ \mathbb{T}}=\mathbb{R} × \mathbb{Z}$
the dual measure on $\widehat{\mathbb{G} }$ is measure on $\mathbb{R} × \mathbb{Z}$.
$$g : \mathbb{R} × \mathbb{Z}→\mathbb{C}\quad by \quadρ(g)= \int_{\mathbb{R}} ∑g(x,n) ~ \mathrm{d}{x}. $$
if μ and ρ are correct measures?