What is a random point in the space of generalized functions $F=\{f|f:\mathbb R\to \mathbb R\}$？

Question

What is a random point in the space of generalized functions $F=\{f|f:\mathbb R\to \mathbb R , \ \int_\mathbb R f=1 \}$？

In other words, what is a function-valued random variable?

Answer 1

The $F$ in the OP is not what is usually called the space of generalized functions. The latter would usually mean the spaces of Schwartz distributions $\mathcal{D}'(\mathbb{R})$ of $\mathcal{S}'(\mathbb{R})$. These have a natural topology called the strong topology and thus a Borel sigma algebra $\mathcal{B}$. A random element in these spaces is a distribution valued random variable $T$. Namely it is a $(\mathcal{F},\mathcal{B})$-measurable map $T:\Omega\rightarrow \mathcal{D}'(\mathbb{R})\ {\rm or}\ \mathcal{S}'(\mathbb{R})$, where $(\Omega,\mathcal{F},\mathbb{P})$ is some probability space.