I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect.
The background is that I have proven a theorem containing an integral over a real-valued function on a compact interval. I then noticed that, in the proof, I only used the following properties:
- Integral of a constant function: $\int_a^b c\,\text{d}x = (b - a) c$ for any non-negative real number $c$ and $a\leq b$ .
- Integrability on subintervals: if $a\leq a'\leq b'\leq b$ and $f:[a;b]\to\mathbb{R}$ is integrable on $[a;b]$, it is also integrable on $[a';b']$ .
- Monotonicity: if $a\leq b$ and $f,g:[a;b]\to\mathbb{R}$ are integrable on $[a;b]$ and $f(x)\leq g(x)$ for all $x\in[a;b]$, then $\int_a^b f(x)\,\text{d}x \leq \int_a^b g(x)\,\text{d}x$ .
- Splitting an integral: if $a \leq b \leq c$ and $f:[a;c]\to\mathbb{R}$ is integrable on $[a;c]$, then $\int_a^c f(x)\,\text{d}x = \int_a^b f(x)\,\text{d}x + \int_b^c f(x)\,\text{d}x$ .
Now, I think that any reasonable concept of integration (Riemann, Lebesgue, Kurzweil–Henstock, …) should fulfil these four properties. But the interesting question is: are there any concepts other than integration (i.e. things that are not merely a generalisation of the Riemann integral) that fulfil these properties? If yes, I could generalise my theorem to this wider class of concepts.
The kinds of things I considered are something like $$\text{‘given}\ f\ \text{and}\ [a;b]\text{,}\ \text{return}\ (b-a)\cdot \sup\limits_{x\in[a;b]} f(x)\text{’}$$ where any continuous function is considered integrable. However, this violates the ‘splitting’ property.
Are there any interesting ways -- other than integration -- to turn a function $f$ on a compact interval $[a;b]$ into a single real number in a way that fulfils the above four properties?
