I encounter an integral in the following form $$I=\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}(2\pi)^4\delta^{(4)}(p-k_1-k_2) f(k_1)g(k_2).\tag{1}$$ And surprisingly I found the following two procedures lead to different results. The first one is obtained by integrating over $k_1$ and relabeling $k_2$ as $k$ $$I_1=\int\frac{d^4k}{(2\pi)^4}f(p-k)g(k).\tag{2}$$ The second one is obtained by integrating over $k_2$ and relabling $k_1$ as $k$ $$I_2=\int\frac{d^4 k}{(2\pi)^4}f(k)g(p-k).\tag{3}$$
Usually, $I_1$ is equal to $I_2$ because one can take the variable transformation $k\rightarrow p-k$. However, this is not allowed if the integral need to performed with a regulator $\Lambda$ and then let $\Lambda\rightarrow\infty$. (Is this called Cauchy principal value?)
[Just to show in what case a regulator is needed, I will give what I encountered. In my case, we have $$g(k)=2\pi\, {\rm sgn}(k_0)\delta(k_0^2-\vec{k}^2-m^2)\\ f(k)=(1/2+f_B(k_0))2\pi\, {\rm sgn}(k_0)\delta(k_0^2-\vec{k}^2-m^2),$$ where $f_B(k_0)=1/(e^{\beta k_0}-1)$. For those who are familiar with thermal field theory, one shall recognise the integral as the one-loop self-energy for certain scalar field theory. Then one can integrate over k_0 first. The remaining integral can be carried out using the spherical variables $|\vec{k}|, \theta, \phi$. One finds that the integral over $|\vec{k}|$ is ill-defined. But we can first let the maxmum of $|\vec{k}|$ be $\Lambda$ and then let $\Lambda\rightarrow \infty$ finally.]
Now my question is, when this happens, what is the correct result for the integral (1)?