I asked this question on math.stackexchange.com, but haven't received an answer, so I thought I'd brave the waters here.

Suppose $M$ is a smooth oriented compact connected $m$-dimensional manifold and let $A^m(M)$ denote the set of smooth exterior differential $m$-forms on $M$. We can integrate members of $A^m(M)$, and the mapping $\phi\mapsto\int_M\phi$ has several properties:

- It's not identically $0$.
- It's linear.
- It's symmetric.

By "it's symmetric", I mean that if $f:M\to M$ is an orientation-preserving diffeomorphism, then $\int_M \phi = \int_M f^*\circ \phi$. I think that expresses the idea of symmetry I have in mind. Basically, all points on $M$ are equivalent as far as integration is concerned.

My question is: Do these properties uniquely determine the space $A^m(M)$ and the usual definition of integration? More precisely, is there some smooth compact connected $m$-manifold $M$, a space $S$ of smooth sections of the tensor bundle of $M$, a large family $F$ of diffeomorphisms of $M$, and a non-trivial linear functional $\lambda:S\to\mathbb{R}$ such that $\lambda( \phi) = \lambda( f^*\circ \phi)$ for all $\phi\in S$ and $f\in F$, but is not just integration of exterior $m$-forms, up to constant scale factor?

My motivation for asking this question: I suspect almost everyone who's ever learned about integration of differential forms on manifolds has wondered why those particular definitions were chosen. The answer is probably some variant of, "Because those definitions work." But must it be this way? Are they the *only* definitions that make integration work?