Delta function of two variables in integral of three variables

Question

I want to integrate a function like:

$$ Z = \int\int\int dt dt' dt'' \, F(t'',t)G(t,t') \delta(t-t')$$

where $F$ and $G$ are arbitrary functions.

Clearly we must have $G(t,t)$ in the expression due to the $\delta$ function, but I'm not sure what to do with $F(t'',t)$. How is $Z$ ultimately affected by the $\delta$ function? Do we just end up with:

$$ Z = F(t',t')G(t',t')$$

Answer 1

The delta function will go away once you integrate in $t$ or $t'$, leaving behind two ordinary integrals. If, say, you integrate in $t$ first, you get

$$Z=\iint dt' dt'' F(t'',t') G(t',t')$$

which no longer has a delta function in it.