On the foundations of abstract Riemannian integration In my real analysis course, we have tackled the theory of integration (according to Riemann) from an abstract perspective, which could be summarized in the following way:


*

*Define an elementary measure space $(A,\mathcal E,m)$ consisting of a set $A$, a family of subsets of $A$ that has the algebraic structure of a semiring, and a finitely additive measure $m : \mathcal E \to [0,+\infty)$;

*Define $s:A\to\mathbb R$ to be a step-function over $A$ when there is a finite partition $\{E_1,\dots,E_p\}\subset\mathcal E$ of $A$ such that, for some $c_k \in \mathbb R$, we have
$$s(x) = \begin{cases}
c_k &  x\in E_k,\ k\in\{1,\dots,p\} \\
0 & x\notin A
\end{cases} $$
and its integral to be
$$\int_A s(x)dm = \sum_{k=1}^p c_k m(E_k) $$

*Given a function $f:A\to\mathbb R$, define $\mathcal I_\pm(f)$ as follows: $r_\pm \in \mathcal I_\pm(f)$ iff there exist $s_\pm$ such that $\forall x \in A$ we have $s_-(x)\leq f(x)\leq s_+(x)$ and $\int_A s_\pm(x)dm = r_\pm$;

*The integral of $f$ would then be
$$\int_A f(x)dm = \inf \mathcal I_+(f) = \sup\mathcal I_-(f) $$
whenever $\mathcal I _\pm(f) \neq \varnothing$, the two bounds exist and are equal.


My question concerns the reasons for choosing the specific algebraic structure of a semiring for the family of subsets of $A$ at the very beginning.

Would it be possible to construct an alternative (similar?) theory of integration based on other (simpler or more complex) algebraic structures, e.g. groups of sets, fields of sets...

And a more general question:

How does the overall organization of the theory of integrals depend on which algebraic structure we choose at the beginning? How do the fundamental results (e.g. FTC) change?


EDIT: Update concerning the naming of measure-theoretic semirings. Consider the result below and the subsequent remarks (translating from my textbook!):

Proposition. Let A be a nonempty set and $\mathcal E$ be a semiring of subsets of A. Also, let $E_1,\dots,E_p \in \mathcal E$. Then it is possible to represent the union of the family $\{E_1,\dots,E_p\}$ as the union of a family of pairwise disjoint elements of $\mathcal E$.
  Secondly, given another finite family of elements of $\mathcal E$, $\{E_1',\dots,E_q'\}$, then it is possible to represent the difference $$\left(\bigcup_{k=1}^p E_k\right)\setminus\left(\bigcup_{j=1}^qE_j'\right) $$
  as the union of a finite family of pairwise disjoint elements of $\mathcal E$.

Remark. The result above allows us to explain the origin of the name semiring of subsets. Let $\mathcal E$ be a semiring of subsets of $A$. Consider the family $\mathcal R$ of all subsets of $A$ that can be represented as finite unions of pairwise disjoint elements of $\mathcal E$. Then, because of the Proposition, we see that if two sets are in $\mathcal R$, their union, intersection and difference also belong to $\mathcal R$. In particular, it makes sense to consider the two applications from $\mathcal R \times \mathcal R$ to $\mathcal R$ that map each couple $(X,Y)\in\mathcal R\times\mathcal R$, respectively, to $(X\setminus Y)\cup(Y\setminus X)$ and $X\cap Y$. These are therefore operations in $\mathcal R$ and one can show that, if they are adopted as additive and multiplicative operation, $\mathcal R$ becomes a ring in the sense of abstract algebra, hence the name semiring for the family $\mathcal E$ we started with. More precisely, if we transfer the sum and product of the ring $\mathbb Z_2$ to the set $\mathcal R'$ of all functions $A \to \mathbb Z_2$, then $\mathcal R'$ becomes a ring and the ring $\mathcal R$ we'd built is isomorphic to a subring of $\mathcal R'$, the isomorphism being as follows: to each $X \in \mathcal R$ we associate the function of $\mathcal R'$ that evaluates to $1$ in $X$ and to $0$ in $A\setminus X$.
 A: A semiring of sets is not the same thing as a semiring in the algebraic sense.


*

*An algebraic semiring is a set $S$ together with two binary operations, $\oplus$ and $\otimes$, satisfying some algebraic properties -  e.g. $\oplus$ and $\otimes$ are associative, $\oplus$ is commutative, etc.

*A measure-theoretic semiring is a collection of subsets $\mathcal{E}$ of some "base set" $X$ satisfying some combinatorial properties, e.g. closure under intersection.
Things get worse if we look at the term "field of sets." As far as I'm aware, there is no connection between fields of sets and algebraic fields, at all.
So I think the generalization you're looking for should not be expected; this is just a coincidence of terminology.

Incidentally, it's worth pointing out that several other strengthenings of "semiring of sets" - $\sigma$-ring, $\sigma$-field, etc. - are fundamentally non-algebraic: they are defined in terms of infinitary operations (e.g. countable intersections). So these systems should really be thought of as combinatorial, not algebraic, objects.
