Let $C^\infty(\mathbb{R})_{int}$ denote the subspace of $C^\infty(\mathbb{R})$ whose elements are integrable on $[0,\infty)$, and let $C^\infty(\mathbb{R})_{int}^0$ denote the codimension one subspace of functions whose integral is zero. Here's a rephrasing of the desired properties of $A$ and $T$:
• The restriction of $T$ to integrable (on $\mathbb{R}_{\geq 0}$) smooth functions C^\infty(\mathbb{R})_{int}$lands in a distinguished subspace$\mathbb{R} \subset A$, and is given by ordinary integration. • As far as I can tell, there are lots of subspaces of [Edit:] Let$C^\infty(\mathbb{R})$that are closed under addition X$ is a space of integrable smooth functions , and are closed under the action of addition by $C^\infty(\mathbb{R})_{int}$, such that $\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$. Indeed, essentially any asymptotic growth class yields such a space. Taking mathbb{R}^\times_{>0}$acts freely on the span of quotient vector space$X/C^\infty(\mathbb{R})_{int}$. If a set universal target$A$for integration existed, then$X/C^\infty(\mathbb{R})_{int}^0$should admit an injection to$A$, because your list of such spaces yields conditions specifies no new further relations, so . The problem (as pointed out by Tao) is that there are lots of smooth functions with nontrivial stabilizer in$\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$. I think a universal recipient common method to removing such a difficulty is to ignore the requirement that integration be$A$\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$-equivariant. Then your universal space is necessarily much larger than just $\mathbb{R}$.C^\infty(\mathbb{R})/C^\infty(\mathbb{R})_{int}^0$. 1 I don't see why you want$A$to be an algebra, since the integral of 1 doesn't seem like a reasonable unit. Did you want some compatibility with higher dimensional integrals using the Fubini theorem? Otherwise, if you follow Kähler's lead, it seems more natural to expect a real (or complex) vector space. Here's a rephrasing of the desired properties of$A$and$T$: 1. Linearity of$T$. 2. The restriction of$T$to integrable (on$\mathbb{R}_{\geq 0}$) smooth functions lands in a distinguished subspace$\mathbb{R} \subset A$, and is given by ordinary integration. 3. Good behavior under the action of the group$\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$generated by translations and orientation-preserving dilations. As far as I can tell, there are lots of subspaces of$C^\infty(\mathbb{R})$that are closed under addition of integrable smooth functions, and are closed under the action of$\mathbb{R} \rtimes \mathbb{R}^\times_{>0}$. Indeed, essentially any asymptotic growth class yields such a space. Taking the span of a set of such spaces yields no new relations, so a universal recipient$A$is necessarily much larger than$\mathbb{R}\$.