I have been reading the article "[A peak set of Hausdorff dimension $2n-1$ for the algebra $A(\mathcal{D})$ in the boundary of a domain $\mathcal{D}$ with $C^\infty$-boundary in $\mathbb{C}^n$][1]" by B. S. Henriksen. It is a short paper but quite dense in my opinion. 

But I have a problem with some of the definitions. For example in the begining of page 275 there are three definitions that to me look circular. I have not seen anything similar before. Does anyone has a better understanding of this part ?  [![enter image description here][2]][2]




  [1]: https://link.springer.com/article/10.1007/BF01457313
  [2]: https://i.sstatic.net/2VBLT.jpg