Seeking Name of Theory for multiple integral Consider two definite integrals:
\begin{equation}
I_1 = \int_{R_1} f(x) \:dx\qquad I_2 = \int_{R_2} g(y) \:dy
\end{equation}
Then, 
\begin{equation}
 I_1 \cdot I_2 = \left[ \int_{R_1} f(x) \:dx \right] \left[  \int_{R_2} g(y) \:dy\right] 
\end{equation}
Which under certain conditions becomes:
\begin{equation}
I_1 I_2 =  \int_{R_1} \int_{R_2} f(x) g(y)  \:dx \:dy
\end{equation}
I'm unsure whether I have to force the following, but here $f(x), g(y)$ are Real Valued Functions continuous on $R_1$ and $R_2$ respectively. 
My question is: What theorem(s) is required in order for this property to hold? i.e. how do we know when we can combine and separate multiple integrals? 

This is just linearity of the integral.  It's automatic as long as the product $I_1\cdot I_2$ itself is meaningful (i.e. we need $I_1, I_2, I_1I_2$ to exist in some sense, so we shouldn't have e.g. $I_1=0, I_2 = \infty$)
Edit: After thinking a little more, it occurs to me that we also need the "inner" integral in your iterated integral to be finite.
