##### Department of Mathematics,

University of California San Diego

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### Math 248 - Real Analysis Seminar

## Adam Bowers

#### University of Seattle

## Multilinear Measure Theory and the Radon-Nikodym Theorem

##### Abstract:

The origins of multilinear measure theory (also known as multidimensional measure theory) can be traced back to the work of Fr$\mathrm{\acute{e}}$chet in 1915. Fr$\mathrm{\acute{e}}$chet characterized the bounded bilinear functionals on $C[0, 1]$, generalizing the characterization of bounded linear functionals given by Riesz. It was not until much later that these bounded bilinear functionals came to be identified with set functions called bimeasures. Since that time, multilinear measure theory has developed, and contains many interesting and deep results. Despite the progress, however, several key measure-theoretic results have eluded satisfactory generalization. In this talk, we will use results in operator theory to provide a generalization of the Radon-Nikod$\mathrm{\acute{y}}$m theorem, and then use it to prove a bounded convergence theorem for bimeasures.

Host: John Eggers

### March 31, 2011

### 11:00 AM

### AP&M 7321

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