Consider zero function on $\mathbb{R}$. If we use Lebesgue integral this will be $0\cdot\infty$. How is it defined? In analysis, we have to work it out with infinitesimals, so it is not necessarily zero.
Also on the other hand, if we integrate infinity on a null set the result is also zero. But what if I integrate Dirac function / Dirac comb?
On
use the Lebesgue integral for non-negative functions, which uses $0⋅∞=0$ as a convention, same argument for if you're integrating the function that is infinity on a null set.
It is not correct to have two measures in an integral, $$ \int_{\mathbb R} \sum_n \delta (x+n) d\mu=?$$
If you were instead computing $$ \int_{\mathbb R} \sum_n \delta (x+n) dx$$ then [the $dx$ is just a notation, and] you should note that the null sets with respect to the Dirac comb measure are the sets that avoid the integers.
The dirac function is the measure given by $$ \int_U \delta (x) dx = \begin{cases} 1 & 0 \in U \\ 0 &\text{otherwise}\end{cases}$$ To define the dirac comb from this, you will also need to define translations of measures, addition of measures, and some appropriate space to take an infinite sum in. Alternatively you can just define it as the measure $$\int_U \sum_n \delta (x+n)dx = | \mathbb Z \cap U|$$
I think we define this quantity, for any set of infinite measure, to be $0$ (and I just consulted Bass's Real Analysis for Graduate Students, Ch.6, which agrees with me). This is convenient because we want examples like yours to come out to $0$.