The Hahn-Banach Theorem implies the existence of a non-Lebesgue measurable set
Foreman, Matthew ; Wehrung, Friedrich
HAL, hal-00004713 / Harvested from HAL
Text on HAL
In this paper we present a new way for proving the existence of non-measurable sets using a convenient operation of a discrete group on the Euclidian sphere. The only choice assumption used in this construction is the Hahn-Banach theorem, a weaker hypothesis than the Boolean Prime Ideal Theorem. Our construction proves that the Hahn-Banach theorem implies the existence of a non-Lebesgue-measurable set of reals. In fact we prove (under Hahn-Banach theorem) that there is no finitely additive, rotation invariant extension of Lebesgue measure to all subsets of the three-dimensional Euclidean space.
Publié le : 1991-07-05
Classification:  amenable group,  sphere,  Hahn-Banach Theorem,  discrete group,  43A07, 28A20, 28A60, 28A99, 28B10, 03E25,  [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] 
@article{hal-00004713,
     author = {Foreman, Matthew and Wehrung, Friedrich},
     title = {The Hahn-Banach Theorem implies the existence of a non-Lebesgue measurable set},
     journal = {HAL},
     volume = {1991},
     number = {0},
     year = {1991},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00004713}
}

Foreman, Matthew; Wehrung, Friedrich. The Hahn-Banach Theorem implies the existence of a non-Lebesgue measurable set. HAL, Tome 1991 (1991) no. 0, . http://gdmltest.u-ga.fr/item/hal-00004713/