Partial group algebra
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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 (X, d, μ) be a metric probability space; that is, let (X, d) be a metric space and let μ be a probability measure on the Borel sets of (X, d). The Laplace functional of (X, d, μ) is the function
defined by
Properties
The Laplace functional of (X, d, μ) can be used to bound the concentration function of (X, d, μ). Recall that the concentration function of (X, d, μ) is defined for r > 0 by
where
In this notation,
References
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