Talagrand's proof of the sufficiency of existence of a majorizing measure for the sample boundedness of processes with bounded increments used a contraction from a certain ultrametric space. We give a short proof of existence of such an ultrametric using admissible sequences of nets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-1-7, author = {Witold Bednorz}, title = {Majorizing Measures and Ultrametric Spaces}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {60}, year = {2012}, pages = {91-100}, zbl = {1262.60034}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-1-7} }
Witold Bednorz. Majorizing Measures and Ultrametric Spaces. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 60 (2012) pp. 91-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-1-7/