So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\mathfrak{p}}$, i.e, $G=\underset{\mathfrak{p}}{\prod}G_{\mathfrak{p}}$ such that $G_{\mathfrak{p}}=H_{\mathfrak{p}}$ for almost all $\mathfrak{p}$, where $H_{\mathfrak{p}}$ are compact and open subsets of $G_{\mathfrak{p}}$. Now when defining the integration of functions over $G$, we consider functions on $G_{S}$, which are subgroups of G, where S is any finite indexed set, such that whenever $\mathfrak{p}\not\in S$, we have $G_{\mathfrak{p}} = H_{\mathfrak{p}}$. So for now, just focus on these functions defined on $G_{S}$. When defining the integration over locally compact groups, we select Haar measures such that $\int_{H_{\mathfrak{p}}} d \mu _{\mathfrak{p}}=1$ (we can select this because we have uniqueness of haar measures up to a constant). Consider a function on a $G_{S}$. Tate, in his thesis, only takes functions of the form $f=\underset{\mathfrak{p}}{\prod}f_{\mathfrak{p}}$ such that $f_{\mathfrak{p}}=1$ on $H_{\mathfrak{p}}$ for almost all $\mathfrak{p}$. And then we perform integration over these functions, which are fairly easy, as now we just have some finite product of functions and measures and all other factors just run down to 1.
My question is, what about analysis on general functions. What if we didn't have any conditions on this $f$. Then it seems non-trivial to even take advantage of such restricted products. Is there a theory for analysis on such general functions $f$. I'm guessing there will be, it would be great if you could point out to some sources regarding this.
Thanks!