The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated here :
How to construct a continuous finite additive measure on the natural numbers
Other elements also appear there :
Thank you for your time.
It is consistent with ZF that there is no nonprincipal (i.e., singletons have measure 0) finitely additive probability measure at all. This appears in Definability of measures and ultrafilters by Pincus and Solovay [J. Symbolic Logic 42 (1977), no. 2, 179–190, doi: 10.2307/2272118, JSTOR], but the exact attribution is complex. From the intro to that paper:
Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo-Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF+ DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF+ DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.
