Unifying different theories of integration

Question

Having a few theories of integration in mind (not just Riemann and Lebesgue, see also Darboux and Cauchy integral defined here), I am thinking about how those integration theories are related. To do this, let me invent the following definition.

Definition. An integration theory on an open set $X\subseteq\mathbb R$ is a tuple $(E,T)$ where $E$ is a subspace of the vecctor space $\mathbb R^X$ (Space of all real-valued functions on $X$) and $T$ is a linear functional $E\to \mathbb R$, satisfying the following properties:

1. The indicator function of every open/closed/half-open interval ($\mathbf 1_{(a,b)},\mathbf 1_{(a,b]}$ and so on) are in $E$.
2. $T(\mathbf 1_{(a,b)})=b-a$.
3. $T$ is continuous with respect to the supremum norm. In other words, $Tf_n\to Tf$ whenever $f_n$ converges uniformly to $f$ in $X$. This is, of course, only need to be true for $f\in E$.

This definition is not necessarily satisfactory because it is not immediate that some other important properties of integrals are guaranteed. For example, Riemann and Lebesgue integrals defines $L^p$ seminorms on certain class of functions, but I am not sure if every "integration theory" in the sense described above defines a norm. For the moment, let assume that an integration theory defines an $L^p$ seminorm (possibly after refining the definition).

Let's further define $(F,U)$ to be a sub-theory if $F$ is a subspace of $E$ and $U=T$ on $F$. Or we can say $(E,T)$ is an extension of $(F,U)$. So Riemann's theory is a subtheory of Lebesgue's theory. We can say an extension is complete if the norm makes $E$ a complete metric space.

Note that each integration theory is associated with an $L^p$ seminorm. If we put those norms together, we can analyse whether one theory is dense in another, and things like that.

I am sorry if this question is too vague. The questions is: are we going to get some useful results if we continue this way of thinking? Are there any similar theories that unifies different types of integrals (Lebesgue, Riemann, etc) in the sense described above?