Chapter 3 of the deeplearningbook gives this formula (equation_1) for "Empirical distribution"
\begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation}
Wiki gives this formula (equation_2) for "Empirical distribution"
$$\widehat {F}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}$$
where ${\displaystyle \mathbf {1} _{A}}$ is the indicator of event A. For a fixed t, the indicator ${\displaystyle \mathbf {1} _{X_{i}\leq t}}$ is a Bernoulli random variable with parameter $p = F(t)$
people on this post says equation_1 and equation_2 are the same. Does that mean $\delta(x - x^{(i)})$ and the indicator ${\displaystyle \mathbf {1} _{X_{i}\leq t}}$ are the same?
The first formula is a probability density, whereas the latter is a comulative distribution function. You can easily check that $\hat F(t)=\int_{-\infty}^t\hat p(x) dx$ by using the fact that the derivative of a step function is the dirac delta.