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This question is a little on the softer and speculative side, so bear with me.

Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable function $f:\Omega \to \Omega'$ is one such that for every $A\in \Sigma '$ we have that $f^{-1}(A)\in \Sigma$.

Let $\Omega'$ be $\mathbb{R}$ with the usual Borel measure. We say that a (let's say probability) measure $\mu$ on $(\Omega, \Sigma)$ acts on a measurable $f:\Omega \to \mathbb{R}$ by $$\mu (f) \equiv \int\limits_{\Omega} f(x) \, d\mu (x) .$$

This is a way to define measures, in many settings — by the way they act on measurable functions.

My question: is there a way to define measures solely through classes (I guess algebras? not sure) of functions? Something like: given a class of functions $\mathcal{F}$ from $\Omega \to \mathbb{R}$, a measure is a functional on $\mathcal{F}$ with such and such properties.

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    $\begingroup$ Isn't this exactly what's done in (one of) the Riesz representation theorem? $\endgroup$
    – LSpice
    Commented May 16, 2023 at 1:44
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    $\begingroup$ The Riesz representation theorem tells you that every positive linear functional on continuous compactly supported functions on a locally compact Hausdorff space $X$ comes from a Radon measure on $X$, and vice versa. So it would seem you can't do better than "positive linear functional" for some classes of $\mathcal{F}$! $\endgroup$
    – Zhen Lin
    Commented May 16, 2023 at 4:01
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    $\begingroup$ This is actually really common in the literature (though maybe not in elementary textbooks). See: Daniell integral. If you e.g. read the opening parts of Federer's Geometric Measure Theory, he discusses such approaches at length. $\endgroup$
    – user378654
    Commented May 16, 2023 at 5:58
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    $\begingroup$ Amir, not sure if this is what you are searching for, but perhaps this book (the very last one) by Constantin Carathéodory, Algebraic theory of measure and integration. 2nd ed. (English) New York: Chelsea pp. 378 (1986), MR0917481, Zbl 0665.28001 is what you need. $\endgroup$
    – Daniele Tampieri
    Commented May 16, 2023 at 6:51
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    $\begingroup$ Actually this was also somehow the approach of Bourbaki, that is measures mainly on topological spaces as linear functionals on spaces of continuous functions. $\endgroup$
    – Pietro Majer
    Commented May 16, 2023 at 7:43

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Yes, that is the approach of Daniell and Stone. To see how the approach works and how general it is, you can take a look at the four-part series on "Notes on Integration" by M. H. Stone. See here, here, here, and here.

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  • $\begingroup$ This are fantastic, thank you. Do you know if this broader sense of integration was ever used (i.e., in cases where it cannot be reduced to "usual" measure and integration)? $\endgroup$
    – Amir Sagiv
    Commented May 25, 2023 at 18:35

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