My precise question,: Is there any measurable function $\omega:\mathbb R\to \mathbb R$ such that for every compactly supported smooth function $\phi \in C_c^\infty(\mathbb R),$ $$ \int_\mathbb R \phi(x) \omega(x) dx = \int_\mathbb R \phi(x) \,\delta(dx):= \phi(0).$$
In other words this might help to show that the weak derivative of the Heaviside function does not exist. Heaviside is defined as follows $$ H(x) = 1 ~~~\text{if}~~~ x>0~~~ \text{and} ~~~0 ~~~\text{elsewhere} $$
The first title of the post contained a question whether or not a Dirac measure is absolutely continuous wrt. the Lebesgue measure. I addressed this question, which is now, after the title change, inactual.
Concerning the question given in the title of the post, the Lebesgue measure of a singleton $\{x\}$ is zero, while its Dirac measure $\delta_x$ is $1$. That is why Dirac measure is not absolutely continuous wrt. the Lebesgue measure.