  Quasi-infinitely divisible (QID) distributions have been recently introduced
by Lindner, Pan and Sato (\textit{Trans.~Amer.~Math.~Soc.}~\textbf{370},
8483-8520 (2018)). A random variable $X$ is QID if and only if there exist two
infinitely divisible (ID) random variables $Y$ and $Z$
s.t.~$X+Y\stackrel{d}{=}Z$ and $Y$ is independent of $X$. In this work, we show
that a family of QID completely random measures (CRMs) is dense in the space of
all CRMs with respect to convergence in distribution. We further demonstrate
that the elements of this family posses a L\'{e}vy-Khintchine formulation and
that there exists a one to one correspondence between their law and certain
characteristic pairs. We prove the same results also for the class of point
processes with independent increments. In the second part of the paper, we show
the relevance of these results in the general Bayesian nonparametric framework
based on CRMs developed by Broderick, Wilson and Jordan (\textit{Bernoulli},
\textbf{24}, 3181-3221 (2018)).
