show/hide this revision's text 2 I said something false.

If you are interested in manifolds, then you might be interested in various related notions of measures and distributions. Let $M$ be a smooth manifold with algebra of functions $C^\infty(M)$. There is a very general notion of a distribution, which is that of any linear function $C^\infty(M) \to \mathbb R$. In this framework, a distribution is a measure if it satisfies a positivity condition, namely that it takes everywhere-nonnegative functions to nonnegative numbers.

Another, different definition of "distribution" corresponds to the distribution bundle on $M$, which is a canonical trivializable line bundle on $M$. It can be presented by gluing data and transition amplitudes as follows. Let $U,V \subseteq M$ with $\phi: U \to \mathbb R^n$ and $\psi: V \to \mathbb R^n$ be coordinate charts, and consider the trivial bundles one-dimensional bundles over $U,V$. We glue them together by giving transition data: if $f$ is a section over $U$ of the trivial bundle, on $U\cap V$ we identify it with the section $f \cdot \left| \det \frac{\partial \phi}{\partial \psi}\right|$ of the trivial bundle over $V$. (When $M$ is oriented, this bundle is the same as the determinant bundle $\wedge^{\operatorname{top}} {\rm T}^*M$; the determinant bundle is always a line bundle, and so its square is trivializable, and has a trivializable square root, which is the distribution line bundle whether $M$ is oriented or not.) Note that the transition functions preserve positivity of the sections, and so the notion of "positive distribution" and so on are well-defined.

Finally, if you are interested in a totally algebraic notion of integration for $\mathbb R^n$, you might be interested in the following observation, which in some form is older but nevertheless deserves to be called an observation of Berezin. Namely, the integral, as a linear map $C^\infty_{\operatorname{compact}}(\mathbb R^n) \to \mathbb R$, is uniquely defined up to scalar multiple by the fact that it vanishes on the images of $\frac{\partial}{\partial x_i} : C^\infty_{\operatorname{compact}}(\mathbb R^n) \to C^\infty_{\operatorname{compact}}(\mathbb R^n)$. Here $C^\infty_{\operatorname{compact}}(\mathbb R^n)$ is the algebra of smooth functions with compact support, and $x_1,\dots,x_n$ are the usual coordinate functions on $\mathbb R^n$. There are many situations in which by naming an algebra "of functions" and some "partial derivatives" you can uniquely (up to scalar) determine an "integral". An example is the algebra of de Rham differential forms on an oriented manifolds $M$, and the "partial derivatives" are the de Rham $d$ and the Lie derivatives for all vector fields on $M$. This uniquely picks out the integral that is zero on non-top forms and integrates top forms over $M$. M$ as a canonical "measure" on the "space" whose "algebra of functions" is the differential forms. This is an example of a "superintegral", and it was to motivate a definition of superintegrals that Berezin made the above observation.
show/hide this revision's text 1

If you are interested in manifolds, then you might be interested in various related notions of measures and distributions. Let $M$ be a smooth manifold with algebra of functions $C^\infty(M)$. There is a very general notion of a distribution, which is that of any linear function $C^\infty(M) \to \mathbb R$. In this framework, a distribution is a measure if it satisfies a positivity condition, namely that it takes everywhere-nonnegative functions to nonnegative numbers.

Another, different definition of "distribution" corresponds to the distribution bundle on $M$, which is a canonical trivializable line bundle on $M$. It can be presented by gluing data and transition amplitudes as follows. Let $U,V \subseteq M$ with $\phi: U \to \mathbb R^n$ and $\psi: V \to \mathbb R^n$ be coordinate charts, and consider the trivial bundles one-dimensional bundles over $U,V$. We glue them together by giving transition data: if $f$ is a section over $U$ of the trivial bundle, on $U\cap V$ we identify it with the section $f \cdot \left| \det \frac{\partial \phi}{\partial \psi}\right|$ of the trivial bundle over $V$. (When $M$ is oriented, this bundle is the same as the determinant bundle $\wedge^{\operatorname{top}} {\rm T}^*M$; the determinant bundle is always a line bundle, and so its square is trivializable, and has a trivializable square root, which is the distribution line bundle whether $M$ is oriented or not.) Note that the transition functions preserve positivity of the sections, and so the notion of "positive distribution" and so on are well-defined.

Finally, if you are interested in a totally algebraic notion of integration for $\mathbb R^n$, you might be interested in the following observation, which in some form is older but nevertheless deserves to be called an observation of Berezin. Namely, the integral, as a linear map $C^\infty_{\operatorname{compact}}(\mathbb R^n) \to \mathbb R$, is uniquely defined up to scalar multiple by the fact that it vanishes on the images of $\frac{\partial}{\partial x_i} : C^\infty_{\operatorname{compact}}(\mathbb R^n) \to C^\infty_{\operatorname{compact}}(\mathbb R^n)$. Here $C^\infty_{\operatorname{compact}}(\mathbb R^n)$ is the algebra of smooth functions with compact support, and $x_1,\dots,x_n$ are the usual coordinate functions on $\mathbb R^n$. There are many situations in which by naming an algebra "of functions" and some "partial derivatives" you can uniquely (up to scalar) determine an "integral". An example is the algebra of de Rham differential forms on an oriented manifolds $M$, and the "partial derivatives" are the de Rham $d$ and the Lie derivatives for all vector fields on $M$. This uniquely picks out the integral that is zero on non-top forms and integrates top forms over $M$. This is an example of a "superintegral", and it was to motivate a definition of superintegrals that Berezin made the above observation.