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The subject of this book is closely related to and expands classical integral geometry. In its most advanced areas it merges with those topics in geometrical probability which are now known as stochastic geometry. By the application of a number of powerful yet simple new ideas, the book makes a sophisticated field accessible to readers with just a modest mathematical background.
Traditionally, integral geometry considers only finite sets of geometrical elements (lines, planes etc.) and measures in the spaces of such sets. In the spirit of the Erlangen program, these measures should be invariant with respect to an appropriate group acting in basic space – to ensure that we are still in the domain of geometry. Assume that the basic space is ℝn (as is the case in the most of this book). If the group contains translations of ℝn, then the measures in question are necessarily totally infinite and cannot be normalized to become probability measures. Yet a step towards countably infinite sets of geometrical elements changes the situation: spaces of such sets admit probability measures which are invariant and these measures are numerous.
The step from finite sets to countably infinite sets directly transfers an integral geometrician into the domain of probability. The vast field of inquiry that opens up surely deserves attention by virtue of the mathematical elegance of its problems and as a potentially rich source of models for applied sciences.
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