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In the Book Karatzas,Shreve: Brownian Motion and Stochastic Calculus they gave the following statement:

If $X$ is adapted to $\mathcal F_t$ and Y is a modification of $X$, then $Y$ is also adapted to $\mathcal F_t$ provided that $\mathcal F_0$ contains all the P-negligible sets in $\mathcal F$. Note that this requirement is not the same as saying that $\mathcal F_0$ is complete, since some of the P-negligible sets in $\mathcal F$ may not be in the completion of $\mathcal F_0$.

I don't understand why there are some P-negligible sets in $\mathcal F$ which is not contained in the completion of $\mathcal F_0$. According to the completion, $\mathcal F_0$ should contain all subsets of P-Nullset, how can it be that there are still P-Nullset outside of the completion.

Additionally, why do we require a filtration to be complete? What would happen, if the filtration is not complete?

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    It is possible e.g. that $\mathcal F_0 = \{\varnothing, \Omega\}$. (It is complete!) However, any non-empty $P$-null set is outside of $\mathcal F_0$. – 2017-02-24
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    @zhoraster ok thanks, I got your point. – 2017-02-24

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