The normal writing of sifting property in signal processing is
\begin{equation} \psi\left(x\right) = \int_{-\infty}^{\infty}\delta\left(x - y\right)\psi\left(y\right)\mathrm{d}y, \end{equation} where $\delta$ is defined as \begin{equation} \delta\left(x\right) = \begin{cases} \infty,\ x = 0\\ 0,\ \mathrm{otherwise} \end{cases} \end{equation} and \begin{equation} \int_{-\infty}^{\infty}\delta\left(x\right)\mathrm{d}x = 1. \end{equation}
Then we have a more rigorous definition for $\delta$ as a distribution (functional) defined on the Schwartz space $\mathcal{S}$: \begin{equation} \forall \phi \in \mathcal{S}, \langle \delta, \phi \rangle = \phi\left(0\right). \end{equation} By defining convolution as an operation $*: \mathcal{S} \times \mathcal{D} \to \mathcal{D}$ ($\mathcal{D}$ to be the space of tempered distributions on $\mathcal{S}$): \begin{equation} \langle\psi * T, \phi\rangle = \langle T, \psi^{-} * \phi \rangle , \end{equation} we have \begin{equation} \psi * \delta = \psi. \end{equation} Then my question is, how to relate the above equation which is more rigorous to the one less rigorous typically used in engineering writing?