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Is there an axiomatic characterization of the Lebesgue integral w.r.t. some finite measure $\mu:\mathcal{F}\rightarrow[0,\infty)$, for instance as the function $I$ over the set of real-valued, $\mathcal{F}$-measurable functions that satisfies the following axioms:

  1. Linearity ($I(\alpha f + \beta g) = \alpha I(f) + \beta I(g)$)
  2. Positivity (for every $f \geq 0$, $I(f)\geq 0$)
  3. Continuity from below (for every $f, f_1, f_2, \dots$ such that $f_n\uparrow f$, $I(f) = \lim_{n\rightarrow\infty}I(f_n)$)
  4. Consistency with the measure (for every $A \in \mathcal{F}$, $I(\mathbb{1}_A) = \mu(A)$)

I'm not asking whether the above axioms characterize the Lebesgue integral; I'm asking whether there is any set of axioms - be it the four above or another set - that characterizes the Lebesgue integral. For example, I've read here that there is an axiomatic characterization of the Lebesgue integral through linear functionals; I'd be interested in this characterization as well as in any other characterization, ideally 'simple' as in the four axioms above (which may or may not characterize the Lebesgue integral).

Evan Aad
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  • Related: https://math.stackexchange.com/questions/654427/category-theory-and-lebesgue-integration – Matthew Towers May 30 '18 at 08:04
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    https://en.wikipedia.org/wiki/Daniell_integral –  May 30 '18 at 08:08
  • @gabrielecassese: Thanks. The Daniell integral appears to be very close to what I am after. However, if I understand correctly, the axioms of the Daniell integral apply to a limited set of 'elementary functions', and the general integral is constructed from the integral of the elementary functions. I'm looking for an axiomatic system that will apply to all the (possibly bounded) measurable functions, be they elementary or not, without requiring any further construction. – Evan Aad May 30 '18 at 08:25
  • Plus transposition invariant maybe? $I(1_B)=I(1_{x+B})$ for every $x$. I think that is enough to make it a multiple of Lebesgue measure. – drhab May 30 '18 at 08:40
  • @drhab: If I understand correctly, you are thinking about the Lebesgue integral w.r.t. the Lebesuge measure on the real line. If so, isn't your proposed additional axiom implied by my axiom #4? At any rate, I'm looking for axiomatics for the Lebesgue integral over a general measure space; not necessarily the Lebesgue measure over the real line. – Evan Aad May 30 '18 at 08:53
  • In my view there is no essential difference between the integral function $I$ and the measure $\mu$. This because $\mu$ is completely determining for $I$. A characterization of the measure is somehow the same as a characterization of the integral function. Then a special thing for Lebesgue measure is $\lambda(x+B)=\lambda(B)$. I was indeed thinking of the real line. – drhab May 30 '18 at 09:04
  • Are axioms 1,3,4 not enough? 1 + 4 gets you the integral of all simple functions. Since you can approximate any nonnegative measurable function via an increasing sequence of simple functions, 3 gets you the integral of any nonnegative measurable function. Then applying linearity to decompose the integral of any measurable function into the integral of its positive and negative parts gets you the integral of any L1 function. – jet457 Feb 28 '23 at 02:55

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