# Daniell integral of “generalized (of some sort)” functions?

Let $$E$$ be a (Dedekind $$\sigma$$-complete) Riesz space and $$H\subseteq E$$ a subspace. A Daniell integral $$I\colon H\to\mathbb R$$ is defined to be a positive linear functional which is continuous with respect to the convergence of monotone sequences, that is to say, for any decreasing sequence $$(f_n)\in H^{\mathbb N}$$ such that $$\inf_{n\in\mathbb N}f_n=0$$, we have $$\inf_{n\in\mathbb N}I(f_n)=0$$. Under this condition, we can extend $$I$$ to a Daniell integral on a subspace $$L_1(I)$$ such that $$H\subseteq L_1(I)\subseteq E$$. I refer to a short paper about this.

We know that this approach is more-or-less equivalent to that of Lebesgue for "normal" functions on a set. I don't know a precise proposition to support this, but roughly speaking, given a set $$X$$ and let $$E=\mathbb R^X$$, then Daniell integrals $$I\colon H\to\mathbb R$$ satisfying $$H=L_1(I)$$ where $$H\subseteq\mathbb R^X$$ is a Riesz subspace correspond to measures on $$X$$ (the associated $$\sigma$$-algebra is taken to be maximal) in a one-to-one fashion.

In fact, it seems to me that in case of "normal" functions, even the theory of Daniell integral is parallel to the measure theory. For example, extensions of Daniell integral roughly correspond to Carathéodory's extension theorem in measure theory.

Several developments go beyond the understanding of functions as maps from a set valued in, say, $$\mathbb R$$. For example, there is a philosophy of noncommutative geometry. As far as I understand (I am a layman), A $$C^*$$-algebra $$A$$ will be considered as an algebra of real or complex valued "functions" on a "noncommutative space", which cannot be understood as a sub-algebra of $$\mathbb R^X$$ for a set $$X$$. I wonder the state-of-art of Daniell integral of generalized functions of some sort, like those appear in noncommutative geometry.