Completion of a measure space and Null Sets

Question

I have a confusion on the completion of a measure space:

I understand its definition being the addition of subsets of Null Sets to the original measure, but I don't understand is

1.why are these subsets not present a priori 2.where do the "completing" null sets come from?

An example is the fact that the Borel measure is incomplete, but the Lebesgue forms its completion. Is there a concrete example of a Lebesgue measurable set that is not Borel measurable?

Thanks!

Answer 1

There are models of set theory not satisfying AC where all subsets of $\mathbb{R}$ are Borel, so we can't really find an "explicit" example of a Lebesgue measurable non-Borel set.

However we can do some simple examples: let $X=\{1,2,3\}$ be a set with three points, $\Omega=\{\varnothing,\{1,2\},\{3\},X\}$ a $\sigma$-algebra, $\mu(X)=\mu\{3\}=1$, $\mu\{1,2\}=\mu\varnothing=0$.

The completion of $(\Omega,\mu)$ consists of the $\sigma$-algebra $\tilde{\Omega}=\mathcal{P}(X)$, the power set of $X$, and the measure $\tilde{\mu}(A)=\begin{cases}1&\text{, if }3\in A\\0&\text{ otherwise}\end{cases}$.