##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## A.V. Mikhalev

#### Moscow State University, Russia

## A solution of the Riesz-Radon-Fréchet problem of characterization of integrals

##### Abstract:

\footnotesize 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 Rn. After papers of I.Radon, M.Fréchet and F.Hausdorff the problem of characterization of integrals as linear functionals is used to be formulated as the problem of extension of Radon theorem from Rn on 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 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 nonbounded positive radom measures. The next modern period 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.

Host: Efim Zelmanov

### November 15, 2011

### 2:00 PM

### AP&M 7218

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