nonconstructive characterization of definite integral

Question

Is there a collection of properties that characterizes a notion of a definite integral of real-valued functions? The determinant can be uniquely defined using an explicit definition (via cofactor expansion or the Leibniz formula) or by giving a few properties that collectively pin it down.‡ I'm wondering whether it's possible to do the latter for a notion of a definite integral.

I'm trying to come up with a notion of an integral where the Riemann integral and the Lebesgue integral (with orientation, as described below) both qualify as definite integral operators.

I also want to allow "conservative" or "timid" definite integrals that only apply to piecewise polynomials, only apply to piecewise constant functions, &c.

By definite integral, I mean a way of assigning values to expressions of the form (101):

$$ \int_{a}^{b} f(x) dx \tag{101} $$

I am also considering the definite integral as a four-place relation (102) that is a partial function of its first three arguments. For every triplet in $(\mathbb{R} \to \mathbb{R}) \times \mathbb{R} \times \mathbb{R}$, either there is exactly one result or there are no results.

$$ S \in \left(\mathbb{R} \to \mathbb{R}\right) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \tag{102} $$

$$ S(f, a, b, w) \stackrel{\text{def}}{\iff} \int_{a}^{b} f(x) dx = w \tag{103} $$

We can coax the Lebesgue integral into evaluating expressions of this form by negating the result when $a > b$ . I suspect that the Lebesgue integral with orientation, as described in the link, is the most general possible definite integral.

---

Here's my attempt to come up with some properties that we want to demand of a definite integral.

I am not asking whether this particular collection of properties captures what we intuitively want for a notion of what a definite integral is. I am intending it as a good faith "first attempt" that illustrates what kind of collection of properties I am trying to find and to demonstrate that I've thought about the question.

1) linearity (scalar multiplication)

$$ \int_{a}^{b} kf(x) dx = k \int_{a}^{b} f(x) dx $$ $$ S(f, a, b, w) \iff S(kf, a, b, kw) $$

2) linearity (addition)

$$ \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx = \int_{a}^{b} f(x) + g(x) dx $$ $$ S(f, a, b, u) \land S(g, a, b, v) \iff S(f + g, a, b, u + v) $$

3) additivity of integration on intervals

$$ \int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx $$ $$ S(f, a, b, u) \land S(f, b, c, v) \iff S(f, a, c, u + v) $$

4) Signed area of rectangles with unit height.

$$ \int_{a}^{b} 1 dx = b - a $$ $$ S(1, a, b, b - a) $$

5) near functions have similar definite integrals.

let $\lVert f \rVert_{[a, b]}$ be defined as $\sup \left\{ \left|f(x)\right| \;\text{for}\; x \;\text{in}\; [a, b] \right\}$. $\lVert f \rVert_{[a, b]} \in \mathbb{R_{\ge 0}} \cup \{\infty\}$

This property formalizes the idea that if two functions differ by less than $\varepsilon$ at their widest, then the difference in the integrals differ by less than $\varepsilon$ multiplied by the length of the interval.

let $D_{[a, b]}(f, g)$ denote the distance between two functions only considering the interval $[a,b]$. $D(f, g)$ is defined as $\lVert f - g \rVert_{[a,b]}$ .

For all intervals $[a,b]$ where $a \le b$, forall real functions $f$,

$$ \forall \varepsilon > 0 . \forall g \in \mathbb{R} \to \mathbb{R} . D_{[a, b]}(f, g) < \varepsilon \implies \left| \int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx \right| < \varepsilon (b - a) $$

provided that both integrals are defined

6) continuity in first argument.

For all intervals $[a,b]$ where $a \le b$, forall real functions $f$,

$$ \forall \varepsilon > 0 . \exists \delta > 0 . \forall g . D_{[a,b]}(f, g) < \delta \implies \left| \int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx \right| < \varepsilon $$

provided that both integrals are defined

‡ The determinant is uniquely determined by the fact that it a) is multilinear in the columns of its argument, b) is alternating in the columns of its argument, and c) sends the identity matrices to 1.

Answer 1

The usual way this sort of question is addressed is the Riesz representation theorem. There are various different closely related formulations of this theorem (you can find some others at the link above), but here is one for integration on intervals in $\mathbb{R}$. Let $C([a,b])$ denote the vector space of continuous functions $[a,b]\to\mathbb{R}$.

Theorem: Let $I:C([a,b])\to\mathbb{R}$ be a linear map such that $I(f)\geq 0$ for any $f\in C([a,b])$ such that $f(x)\geq 0$ for all $x\in[a,b]$. Then there is a unique finite Borel measure $\mu$ on $[a,b]$ such that $$I(f)=\int_{[a,b]} f\,d\mu$$ for all $f\in C([a,b])$.

Corollary: Let $I:C([a,b])\to\mathbb{R}$ be as above, and assume additionally that $I(1)=b-a$ and $I(f)\geq M(d-c)$ for any $f\in C([a,b])$, $M\in\mathbb{R}$, and $[c,d]\subseteq[a,b]$ such that $f(x)\geq 0$ for all $x\in [a,b]$ and $f(x)\geq M$ for all $x\in [c,d]$. Then $I(f)=\int_{[a,b]} f\,d\lambda$ for all $f\in C([a,b])$, where $\lambda$ is Lebesgue measure.

As a sketch of how to prove the corollary from the theorem, let $\mu$ be as in the theorem and note that the assumptions of the corollary imply $\mu([a,b])=b-a$ and $\mu([c,d])\geq d-c$ for all $[c,d]\subseteq [a,b]$. If we ever had $\mu([c,d])>d-c$ we could conclude that $\mu([a,b])>b-a$ by breaking $[a,b]$ into subintervals, one of which is $[c,d]$, so we must have $\mu([c,d])=d-c$ always. This then implies $\mu$ is Lebesgue measure.