Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f\_1, f\_2 \in L^1$ by
$$
f\_1(x) = f(x/2),
\qquad
f\_2(x) = f((x + 1)/2).
$$
Let $I = \int\_0^1$. Then $I$ is the unique bounded linear functional on $L^1$ satisfying the equations
$$
I(\text{constant function at }1) = 1,
\qquad
I(f) = (I(f\_1) + I(f\_2))/2.
$$
This, then, is a simple characterization of integration.


Presumably this has been known for a really long time -- maybe for a century? -- but I'm having trouble tracking it down in the literature. I simply can't find it anywhere. 


Does anyone know anything about the history of this result?



Who first found this characterization of Lebesgue integration?
I don't know if this is of help, but I have seen this idea for defining integration elsewhere, specifically on pages 10-11 of Reed and Simon's Functional Analysis. It would go something like this: let $S$ be the space of step functions obtained as linear combinations of characteristic functions of half-open intervals $[k/2^n, (k+1)/2^n)$. Then $S$ is dense in the space $C$ of bounded piecewise continuous functions continuous to the right, with respect to sup norm. By the two conditions, the definition of $I$ is uniquely determined on $S$ and defines a bounded linear functional on $S$ with respect to sup norm, so it uniquely extends to a bounded linear functional on $C$. This in fact defines the Riemann integral $I$, and it's pretty much exactly what Reed and Simon do, except they don't restrict to dyadic rational endpoints.) 


(Then, we can go on and define $L^1[0, 1]$ to be the completion of $C$ with respect to the norm given by $f \mapsto I(|f|)$, and since $I$ is a bounded linear functional also with respect to this norm, it extends uniquely to the completion $L^1[0, 1]$.) 


**Edit:**: Ah, Reed and Simon mention that this approach to the Riemann integral can be found especifically in Dieudonné, Foundations of Modern Analysis, or in Loomis and Sternberg, Advanced Calculus, Addison-Wesley, 1968. Perhaps one can consult these for further history. 


