Can the Laplace transform of Dirac delta function be explained in sense of Riemann integral or Lebesgue integral

Question

This is what has been explained in my differential equation lecture. But I have no idea how this explanation can be justified in the sense of Riemann integral or Lebesgue integral (Actually this course is for sophomore and general measure is never mentioned in this lecture.)

I think that the above mathematical equation has two problem in the sense of Riemann integral or Lebesgue integral.

1. Since $\delta_a(t-t_0)$ cannot be dominated by an integral function, we cannot apply Lebesgue's dominated convergence thm, and change the order of integral and limit.

2. Indeed the $\delta(t-t_0)e^{-st}=0$ except for $t=t_0$, the integral value must be zero.

I think that we need general measure (maybe Dirac measure) to rigorously explain $\mathcal L{\delta(t-t_0)}=e^{-st_0}$.

Am I right?