Publisher Summary This chapter presents the result of Shelah and Raisonnier stating that, in the theory ZF & DC, the inequality ω1 ≤ c implies the existence of a non Lebesgue-measurable subset of R. Also, In the theory ZF & DC, the existence of a set Y R with card(Y) =1 implies the existence of a subset of R nonmeasurable in the Lebesgue sense. measurable in C. Let be a rapid filter in (). Then is not -measurable in C. To establish the Shelah–Raisonnier result, a number of preliminary notions and facts are needed. Some auxiliary statements are considered concerning the descriptive structure of certain filters in (ω). All these statements are due to Talagrand.