Henri Lebesgue measure problem and its first solutions

Problem of the existence of an invariant measure was formulated by H.Lebesgue in his dissertation “Integral, length, area,” published in 1902. Attempts to solve this problem have led to the construction of the first examples of immeasurable sets – Vitali, Van Vleck, Hausdorff, Bernstein, Banach–Tarski, etc. as well as a broad discussion on the admissibility of the use of ineffective methods and concepts of the meaning of “determined” and “exist”, which was vital for the formation of different views on the development of set theory.