The question is as in the title. In QFT literature, $\delta^4(0)$ is said to stand for the volume of entire $\mathbb{R}^4$, where $\delta^4(x)$ is the $4-$dimensional delta function.

Or when defining normal-ordered product of free scalar field operator $\phi(x)$ at a coinciding point, they use the limit operation \begin{equation} :\phi^2:(x):=\lim\limits_{x \to y} \Bigl( \phi(x)\phi(y)-iD^+(x-y) \Bigr) \end{equation} where $iD^+(x-y)$ is a causally supported two-point function made from commutator of $\phi$. $iD^+(x-y)$ is a tempered distribution and it is written in standard literature that $iD^+(x-y) \simeq \delta^4(x-y)$ as $x \to y$.

I have been always curious how to make sense of these expressions rigorously. The free scalar field $\phi(x)$ is in fact an operator-valued tempered distribution as well. So, every object written above is in fact a distribution. How should we understand their limits written in the form of ordinary functions?

In summary,

- What does it mathematically mean by "$\delta^4(0)=$ the volume of entire $\mathbb{R}^4$"?
- How should I make rigorous sense of expression like $\lim\limits_{x \to y} \delta^4(x-y)$?

Could anyone please clarify? I think I am essentially curious about defining a multivariate tempered distribution $T(x,y)$ on the thin diagonal $x=y$.