Partial group algebra

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Revision as of 05:51, 30 January 2011 by en>Michael Hardy (References: endash)
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In mathematics — specifically, in probability theory — the Laplace functional of a metric probability space is an extended-real-valued function that is closely connected to the concentration of measure properties of the space.

Definition

Let (Xdμ) be a metric probability space; that is, let (Xd) be a metric space and let μ be a probability measure on the Borel sets of (Xd). The Laplace functional of (Xdμ) is the function

E(X,d,μ):[0,+)[0,+]

defined by

E(X,d,μ)(λ):=sup{Xeλf(x)dμ(x)|f:X is bounded, 1-Lipschitz and has Xf(x)dμ(x)=0}.

Properties

The Laplace functional of (Xdμ) can be used to bound the concentration function of (Xdμ). Recall that the concentration function of (Xdμ) is defined for r > 0 by

α(X,d,μ)(r):=sup{1μ(Ar)AX and μ(A)12},

where

Ar:={xXd(x,A)r}.

In this notation,

α(X,d,μ)(r)infλ0eλr/2E(X,d,μ)(λ).

References