Even regularizing kernel in $L_1$

Question

I am learning the text book "Optimal transport for applied mathematician" by Filippo Santambrogio.

In the Section 5.1, we try to prove that the Wasserstein distance satisfies the triangle inequality. One way to do it is as follows:

1. We can prove triangle for inequality $W_p(\mu,\nu)$ when $\mu$ and $\nu$ are absolutely continuous measures.
2. Then, we write the triangle inequality for $\mu*\chi_\epsilon$ and $\nu*\chi_\epsilon$, then pass to the limit as $\epsilon\rightarrow 0$.

Here, $\chi_\epsilon$ is called an "even regularizing kernel in $L_1$".

The author didn't define what the "even regularizing kernel in $L_1$" is. My questions are:

1. What is the "even regularizing kernel in $L_1$"?
2. For an arbitrary measure $\mu$, $\mu*\chi_\epsilon$ is guaranteed to be absolutely continuous?
3. Is there any reference for this? I checked some real analysis textbook, but I didn't see this concept.

Thank you so much!