Is there an equivalent to an ECDF with a "<" sign? The empirical cumulative distribution function is defined as
$$
F(x)=\frac{1}{n} \sum_{i=1}^{n} \mathbb I_{x_i \leq x}
$$
Is there an equivalent interpretation of this function as:
$$
F(x)=\frac{1}{n} \sum_{i=1}^{n} \mathbb I_{x_i<x}
$$
The reason I ask is that I have a distribution of non-negative values, around 40% of which are zero. If I evaluate $F(0)$ using the standard ECDF formula I get $F(0)=0.4$. But, for my purposes, I would like to have $F(0)=0$ (i.e., there are no values less than zero).
Does such a function exist?

Remember how the CDF is a right-continuous function? That comes from using the $\le$.
If you use $<$, then you change the function to be left-continuous.
Regarding your exact question about if such a function exists, the answer is that it does, and you wrote out exactly what it is. That function isn’t the classical CDF, but if it has meaning for your application, use it!
Apparently Kolmogorov used the $<$ convention and left-continuity in his first probability book, so you might consider yourself to be in good company by thinking this is reasonable!
