I am just studying a paper by DiPerna and Lions with the title "On the Cauchy problem for Boltzmann equations: global existence and weak stability." You can find it here: http://www.jstor.org/discover/10.2307/1971423?uid=3737528&uid=2&uid=4&sid=21102076873573

If you do not have access to the paper then use this link: http://www.4shared.com/office/eeIqgvxB/MathematicsOn_the_Cauchy_Probl.html

When you go to page 331 you find two Lemmas, Lemma II.1 and Lemma II.2. The first point of Lemma II.1 states that if the collision operator is locally integrable then f is a distributional solution of the Boltzmann equation if and only if f is a renormalized solution of the Boltzmann equation. I do not know how to prove this result and I could not find one in the paper, may you have an idea.

**EDIT**. I just found one prove to Lemma II.1 and II.2 in the Appendix B of the paper: here is a picture: ![Lemma II.1][1]

I do not see why taking delta to 0, this leads to the conclusion that f is a distributional solution, may you can explain me that.