As soon as stochastic geometry deals with structures satisfying invariance properties with respect to some group, such as stationarity or isotropy in Euclidean spaces, there arises the need for a theory allowing averaging with respect to invariant measures. Integral geometry in the sense of Blaschke and Santaló is perfectly made for obtaining such averaging formulas. In this chapter we develop the basic tools, namely intersection formulas for fixed and moving geometric objects, where suitable geometric quantities of the inter-sections are integrated with respect to invariant measures. Basic facts about invariant measures on locally compact topological groups and homogeneous spaces, as far as they are needed for our purposes, are collected in the Appendix in Chapter 13.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Averaging with Invariant Measures. In: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78859-1_5
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DOI: https://doi.org/10.1007/978-3-540-78859-1_5
Publisher Name: Springer, Berlin, Heidelberg
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