A solution of the Riesz-Radon-Fréchet problem of characterization of integrals
Time and place
1 PM on Thursday, February 9th, 2012;
Alexander V. Mikhalev (Moscow State University)
Abstract
The talk is devoted to the problem of characterization of integrals as linear functionals. The main idea goes back to Hadamard. The first well-known results in this field are the F.Riesz theorem (1909) on integral presentation of bounded linear functionals by Riemann-Stiltjes integrals on the segment and the Radon theorem (1913) on integral presentation of bounded linear functionals by Lebesque integrals on a compact in R^n. After papers of I.Radon, M.Fréchet and F.Hausdorff the problem of characterization of integrals as linear functionals is usually formulated as the problem of extension of Radon theorem from R^n to more general topological spaces with Radon measures. This problem turned out to be rather complicated. The history of its solution is long and rich. It is quite natural to call it the Riesz-Radon-Fréchet problem of characterization of integrals. The important stages of its solution are connected with the names of S.Banach (1937-38), Sacks (1937-38), Kakutani (1941), P.Halmos (1950), Hewitt (1952), Edwards (1953), N.Bourbaki (1969), and others. Some essential technical tools were developed by A.D.Alexandrov (1940--43), M.Stone (1948--49), D.Fremlin (1974), and others. In 1997 A.V.Mikhalev and V.K.Zakharov had found a solution of Riesz-Radon-Fréchet problem of characterization for integrals on an arbitrary Hausdorff topological space for unbounded positive Radon measures. The next modern period in the history of this problem for arbitrary Radon measures is connected mostly with results by A.V.Mikhalev, T.V.Rodionov, and V.K.Zakharov. A special attention is paid to algebraic aspects used in the proof.
