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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $E$ be a space with a geometric structure, usually the [[Euclidean space|0XPR1E85W0]] $\\R^d$.\n*Stochastic geometry* is the study of [[random elements|0PNRKMRC55]] defined on $(\\Omega, \\mathcal{F}, P)$ whose values are geometric objects in $E$.\nIts basic objects are:\n\n1. [[point processes|DNDKBVHS3H]], that is, random locally finite sets or counting measures in $E$;\n2. random closed sets, that is, random elements of the family of closed subsets of $E$;\n3. random measures on $E$, and structures built from the above, such as random tessellations, fibre processes and germ-grain models.\n\nThe questions concern the laws of these objects, their mean values such as intensities and volume fractions, and their behaviour under motions of $E$.\n\n## Remarks\n\nExample: the [[Poisson point process|0MTNR28K60]] in $\\R^d$ is a point process, and the [[Boolean model|NK3CZTVEJD]] built from it is a random closed set; both are central models of the field.\n\nNon-example: a fixed lattice $\\Z^d \\subset \\R^d$ involves no randomness, so on its own it is a question of deterministic geometry.\nA randomly shifted lattice $U + \\Z^d$, with $U$ uniform on $[0, 1)^d$, is again a point process and belongs to the field.\n\nThe term geometric probability is often used for the older part of the subject, which deals with a fixed number of random points, lines or bodies.\n",
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