I have great difficulties understanding why the following relation holds:
$$\int^{t}_{0} \int^{t’}_{0}\delta(t_1-t_2)\mathrm{d}t_1\mathrm{d}t_2=\min\{t,t’\}.$$
Our teacher gave us an explanation involving geometry which I am unable to reproduce. I tried to used the integral representation of the delta function but get non convergent results. Thank you for your help!
Integrate out $t_1$ first to get $\int_0^t[t_2\in(0,\,t^\prime)] dt_2$, where for any proposition $p$ the Iverson bracket $[p]$ is $1$ if $p$ is true or $0$ if $p$ is false. This single integral is $\int_{[0,\,t]\cap[0,\,t^\prime]}1 dt_2=\min\{t,\,t^\prime\}$.