absolute continuity - closed subsets sufficient In a paper I currently work with the author wants to show that a complex borel measure on $\mathbb T$ is absolutely continuous. However he only does consider closed null sets of T. Why is this sufficient?
Kind regards, Sebastian

Surely the measure $\mu$ you are considering is Radon. Therefore, it has the following "inner regularity" property: if $A\subset \mathbb T$ is any measurable set, then 
$$\mu(A)=\sup\{\mu(K)\ :\ K\subset A,\ K\ \text{is compact}\}.$$
Since the torus is compact, you can therefore approximate every measurable set with a closed set. 
