Wikipedia defines a $n$-dimensional delta function as just taking
$$\delta (\mathbf{x}) = \prod_{j=1}^n \delta (x_j)$$
but also cautions
However, despite widespread use in engineering contexts, [it] should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.
Wikipedia cites a couple of books, neither of which do I have access to. I am having difficulty imagining what these narrow circumstances would be, and I would like to understand what they are before I tackle a problem that requires this product measure.
Under what (narrow) circumstances is the product measure defined?