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.

everypositive 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 $\mathcal{F}$! $\endgroup$Algebraic theory of measure and integration.2nd ed. (English) New York: Chelsea pp. 378 (1986), MR0917481, Zbl 0665.28001 is what you need. $\endgroup$2more comments