##### Department of Mathematics,

University of California San Diego

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### Functional Analysis Seminar

## Dan Wulbert

#### UCSD

## A Unified Liapanouv Theorem

##### Abstract:

Let $\mu_1, \mu_2, ... , \mu_m$ be non-atomic probability measures on a measurable space $(X, \Sigma)$. Theorem (Liapanouv 1940) ${\mu(\cup) = (\mu_1(\cup) + \mu_2(\cup) + ... + \mu_m(\cup) ): \cup \ {\text{in}}\ \Sigma }$ is a compact convex set. If in addition there is a topology on X and ‚$\Sigma$ is the Borel sets (or the Baire sets, respectively) we can ask when the range of the vector valued measure $\mu$ is obtained even when the measure is restricted to the sets $\cup$ which are open (or the support of a non-negative continuous function, resp.). We will give a couple applications of the Classical Theorem. We will then cast the Liapanouv Theorem in an equivalent form about the range of a vector of integrals on X. In that form we will give a single theorem that, in addition to proving the Classical Liapanouv Theorem, also characterizes when the open sets (or the supports of continuous functions, resp.) suffice. That is let L be a cone of functions. Let S be the supports of functions in L, and let ‚$\Sigma$ be the sigma-algebra generated by S. The three cases above result when $L = L\infty$, the upper-semi-continuous functions on X, and C(X) respectively.

### December 11, 2007

### 12:00 PM

### AP&M B412

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