A probability space $(\Omega, \mathcal{A}, \mathbb{P})$ is called complete iff every subset $\tilde{N}$ of a set $N$ of measure $0$ is measurable, i.e. if $\mathbb{P}(N) = 0$ and $\tilde{N} \subseteq N$ implies that $\tilde{N}\in \mathcal{A}$. Similarly, a filtration $(\mathcal{A}_t)_{t \in I}$ on $(\Omega, \mathcal{A}, \mathbb{P})$ is called complete iff, for all $t \in I$, the space $(\Omega, \mathcal{A}_t, \mathbb{P} )$ is complete.
I have noticed, that completness of the probability space and completeness of filtrations is a frequent assumption in various textbooks on stochastic calculus and Brownian motion. Why is this the case? What is the advantage if we only restrict ourselves to complete measure spaces and filtrations in the study of stochastic processes?
Kind regards, Joker