 He received the USSR State Prize in 1951 and was elected to the position of corresponding member of the Russian Academy of Sciences in 1953.His construction of a Fourier series with non-zero coefficients which converges to zero almost everywhere gave rise to the theory of Menshov sets.He proved the Rademacher–Menchov theorem, the Looman–Menchoff theorem, and the Lusin–Menchoff theorem.