I am struggling to fully understand the steps in a paper I'm reading. The solution to a Kolmogorov equation is in the form
$$\int_{\tau_0}^{\tau_1}\mathbb{E}\bigg[\mathbb{1}_{0}(N_z)\delta\bigg(\tau - \frac{2}{\bar c}\int_{-L}^{z}N_z^\prime dz^\prime\bigg)\bigg|N_{z = -L} = p\bigg]d\tau$$ where $\mathbb{1}_0$ is the indicator function, $\bar c$ is a positive velocity, $p$ is an integer, $N_{z}$ is a jump Markov process which jumps to $n-1$ or $n+1$ (zero is an absorbing state) with equal probability, $\tau$ is a time, $\delta$ is the Dirac delta function and I'm integrating over the time window $\tau\in[\tau_0,\tau_1]$ and the spatial domain is $z\in[-L,0].$
The paper says that
$$\int_{\tau_0}^{\tau_1}\mathbb{E}\bigg[\mathbb{1}_{0}(N_z)\delta\bigg(\tau - \frac{2}{\bar c}\int_{-L}^{z}N_z^\prime dz^\prime\bigg)\bigg|N_{z = -L} = p\bigg]d\tau \\ = \mathbb{P}\bigg[N_{0} = 0,\frac{2}{\bar c}\int_{-L}^{z}N_z^\prime dz^\prime\in[\tau_0,\tau_1]\bigg|N_{z = -L} = p\bigg]$$
I understand that the expectation of the indicator function is simply the probability, but what has happened to the Dirac delta function here?