Multidimensional integrals that diverge by oscillation It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$) or $f(t) = \sin (at+b)$ (or any finite superposition of such functions, even with incommensurable periods) is $0$; for instance, introduce a suitable kernel $g(t)$ such as $\exp(-t^2)$ and define $\int_{\bf R} f$ to be $\lim_{s \rightarrow \infty} \int_{-\infty}^{\infty} f(t) \: g(t/s) \: dt$. But I have trouble when I try to do this for integrals over ${\bf R}^2$, and things go seriously amiss over ${\bf R}^3$. It seems that all mollifiers fail, basically because the surface measure of bodies in ${\bf R}^d$ is "too big" compared to the interior measure when $d > 1$ (a problem that gets worse and worse as $d$ increases).
What I want is a theory of generalized integration over ${\bf R}^d$ such that compactly supported functions integrate to their usual value, while periodic functions with mean value $0$ integrate to $0$; and I want this theory to apply to a vector space of functions (that is, I want arbitrary linear combinations of finitely many generalized-integrable functions to be generalized-integrable).
Is there (or could there be) such a theory?
 A: There is a concept of definite integrals for distributions which has three of the features you request:
continuous functions of compact support are integrable---even functions which are integrable in the sense of Lebesgue;
the trigonometric functions you describe integrate over the real line to zero;
the space of integrable distributions forms a vector space and the integral is a linear function.
The theory was developed about 50 years ago by the portuguese mathematician J. Sebastião e Silva and is completely elementary, being based on his axiomatic approach.  All of his writings are available online and can easily be found using a search machine.
The  definition is based on two simple facts:
every distribution on the line has a primitive;
the existence of an elementary definition of the concept of the limit of a distribution at a point (which can be $\pm \infty$).
Sebastião e Silva also considered the case of multivariate distributions.
Many of the articles mentioned are in portuguese but there is an english version "Integrals and orders of growth of distributions" in the proceedings of a conference on the theory of distributions (Lisbon, 1964) in which he discusses applications to the Fourier transform of distributions.  This and his other writings are available at the site: www.sebastiaoesilva100anos.org.
Information added as an edit since the system rejected a comment. Sebastião was working on an encyclopaedic monograph which would have given a complete presentation of his theory with applications to mathematical physics but his early passing prevented its completion.
