There are two issues: local regularity and decay at infinity which are in duality via the Fourier transform. Here one needs to introduce distributions primarily in order to allow very bad local regularity as arises for instance in the idealized situation of a point charge or a surface charge distribution etc. In 1D, take a unit positive charge and concentrate it at the origin. The object you need is then the delta function. But if you take two such things of opposite charge and take a suitable limit where you put them both at the origin you get things like dipoles which requires the derivative in the sense of distributions of the delta function which is not a signed measure anymore as in Emilio's answer. You can of course play this game in higher dimension and generate multipoles etc.

Now your problem has to do with decay at infinity rather than local regularity. You need a restricted class of distributions which can be fed the constant test function equal to one. Although you will not find that in most textbooks on distributions (especially the "math for physicists" ones) Laurent Schwartz considered lots of spaces other than $\mathcal{D},\mathcal{D}',\mathcal{S},\mathcal{S}'$. there are also $\mathcal{E},\mathcal{E}',\mathcal{O}_M,\mathcal{O}_M',\mathcal{O}_C,\mathcal{O}_C',\ldots$ The one you need here, I think, is $\mathcal{O}_C'$ also called the space of rapidly decaying distributions.

There are two issues: local regularity and decay at infinity which are in duality via the Fourier transform. Here one needs to introduce distributions primarily in order to allow very bad local regularity as arises for instance in the idealized situation of a point charge or a surface charge distribution etc. In 1D, take a unit positive charge and concentrate it at the origin. The object you need is then the delta function. But if you take two such things of opposite charge and take a suitable limit where you put them both at the origin you get things like dipoles which require the derivative in the sense of distributions of the delta function which is not a signed measure anymore as in Emilio's answer. You can of course play this game in higher dimension and generate multipoles etc.

Now your problem has to do with decay at infinity rather than local regularity. You need a restricted class of distributions which can be fed the constant test function equal to one. Although you will not find that in most textbooks on distributions (especially the "math for physicists" ones) Laurent Schwartz considered lots of spaces other than $\mathcal{D},\mathcal{D}',\mathcal{S},\mathcal{S}'$. there are also $\mathcal{E},\mathcal{E}',\mathcal{O}_M,\mathcal{O}_M',\mathcal{O}_C,\mathcal{O}_C',\ldots$ The one you need here, I think, is $\mathcal{O}_C'$ also called the space of rapidly decaying distributions.