How to define a generalized function $f:\mathbb R\to\mathbb R$ s.t. $f(x)\neq 0\forall x\in \mathbb R$, and $\int_\mathbb R f=1$?
How to generalized this generalized function $F$ to be defined over $C^\infty$ i.e.
$$f:C^\infty\to\mathbb R$$
s.t. $\int_{C^\infty}F=1$. What kind of metric/topology/measure we need to properly equip the function space so that our definition makes sense?
Basically, I want to have a uniform distribution defined over $\mathbb R$, s.t. $f(x)=f(y)\neq 0 \forall x,y\in\mathbb R$. And similarly, $F(f)=F(g)\neq0$,
???????
First: Some (many) generalized functions aren't functions. Asking for $f(x)$ does not makes sense.
In any case, no generalized function required, The Gaussian is $C^\infty$ and $$\int_{\Bbb R}e^{-x^2}dx = \sqrt\pi.$$ Divide by $\sqrt\pi$.
About the second question: better use another name like $F$.
You can define a measure in $C^\infty$ with $$\mu(\{0\}) = 1$$ ($0 =$ the function $0$) and $$\mu(\hbox{anything without 0}) = 0$$. Check yourself what will be $\int_{C^\infty}F\,d\mu.$