Does the zero random variable has a density function?

Question

I have the following question:

Does the random variable $0$ which is always zero has a density function?

I'm somehow thinking about this fact and I'm really insecure I mean one should be able to compute $\Bbb{P}(0\leq B)$ but this depends on the set B doesn't it?

Thanks for your help

Answer 1

As @Mahdi noted, it didn't have a Lebesgue density. It has a mass function, or if you prefer its density is a distribution without a measure. See here & here.

Answer 2

It does not have a density with respect to the Lebesgue measure. However, it has a density with respect to the counting measure, which is basically point mass with mass of one at zero.