Laplace transform of the Dirac delta

Example

Let $a \ge 0$ and let $\delta_a$ be the Dirac delta at $a$, that is, the Dirac measure that gives mass $1$ to the point $a$. The delta is not a function, so its Laplace transform is defined as the Laplace–Stieltjes transform of the measure. The point $0$ is included in the domain of integration: $$\mathcal{L}\{\delta_a\}(s) = \int_{[0, \infty)} e^{-st}\, d\delta_a(t) = e^{-as} \quad (s \in \C).$$ Check: integrating a continuous function against $\delta_a$ evaluates it at $a$, and $t \mapsto e^{-st}$ is continuous. In particular $\mathcal{L}\{\delta_0\} = 1$, with the convention that the lower limit is $0^-$. This is consistent with the Laplace transform of the step $H(t - a)$, which is $e^{-as}/s$. Indeed, the delta is the derivative of the step, and differentiation multiplies the transform by $s$ when the initial value is $0$.