A specific problem in changing the order of integrals. Immersed within the world of Physics (Quantum Theory in particular), I came across the following proof for connecting the starting LHS with the finishing RHS.
$$
\begin{eqnarray}
   \int_{t_0}^tdt_1 \int_{t_0}^{t_1} dt_2V(t_1)V(t_2) &=&
   \int_{t_0}^tdt_2 \int_{t_2}^{t}   dt_1V(t_1)V(t_2)\\
&=&\int_{t_0}^tdt_1 \int_{t_1}^{t}   dt_2V(t_2)V(t_1)\\
&=&\frac{1}{2}\left[\int_{t_0}^tdt_1 \int_{t_1}^{t}   dt_2
 + \int_{t_0}^tdt_2 \int_{t_0}^{t_1} dt_1\right]V(t_1)V(t_2)\\
&=&\frac{1}{2}\int_{t_0}^tdt_1 \int_{t_0}^{t}   dt_2V(t_1)V(t_2)\\
\end{eqnarray}
$$
Unfortunately, I've had limited mathematical training (for mathematical physics) and I don't understand the motivation for all of the steps involved. The second line is achieved from the first by change of variables $t_1 \longleftrightarrow t_2$, but intuitively, how would one get from the LHS 1st to RHS 1st and 2nd to 3rd to 4th? 
Clearly, I'm missing the fundamentals, so all comments would be welcome.
 A: The first line is easiest to see by drawing the domain of integration on the $t_1,t_2$ plane: it is a right triangle. To go from the 2nd to 3rd, observe that the LHS of the first line is equal to the 3rd line, therefore each is equal to the average of the two. To go from the 3rd line to the last line, swap variables on the 2nd term on the 3rd line, and obtain
\begin{eqnarray}
& & \frac{1}{2} \left[ \int_{t_0}^t dt_1 \int_{t_1}^t dt_2 + \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 \right] V(t_1)V(t_2) \\
& = &
\frac{1}{2} \left[ \int_{t_0}^t dt_1 \left( \int_{t_1}^t dt_2 + \int_{t_0}^{t_1} dt_2\right) \right] V(t_1) V(t_2) \\
& = & \frac{1}{2} \int_{t_0}^{t_1} dt_1 \int_{t_0}^t dt_2 V(t_1) V(t_2).
\end{eqnarray}
A: I remember first encountering this sort of integral and wondering why there wasn't a more mechanical way to handle them. 
Then I came across the Iverson bracket. 
We find 
$$\begin{eqnarray*}
\int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2\, V(t_1)V(t_2)
&=& \int_{t_0}^t dt_1 \int_{t_0}^t dt_2 \, V(t_1)V(t_2)[t_2 \leq t_1] \\
&=& \int_{t_0}^t dt_1 \int_{t_0}^t dt_2 \, V(t_1)V(t_2) [t_1 \leq t_2]
    \hspace{5ex}\textrm{(change variables)} \\
&=& \frac{1}{2}
    \int_{t_0}^t dt_1 \int_{t_0}^t dt_2 \, V(t_1)V(t_2)
\left([t_2 \leq t_1] + [t_1 \leq t_2]\right) \\
&=& \frac{1}{2}
    \int_{t_0}^t dt_1 \int_{t_0}^t dt_2 \, V(t_1)V(t_2).
    \hspace{5ex}\textrm{(a.e.)}
\end{eqnarray*}$$
