Boolean random functions

A Boolean function f in Rn is the supremum of upper semi‐continuous random functions f'i which are almost surely positive, bounded with compact support and centred at the Poisson points (i). They generalize to functions of classical Boolean model for sets. The Boolean function f may be studied via its subgraph, i.e. as a random set in Rn x R. The key notion is then the functional Q(Bt), i.e. the probability that a compact set Bt centred at altitude t misses the subgraph of f. The general expression of Q(Bt) is given, and followed by a series of important derivations (volumes, gradients, numbers of summits, etc). Theorems of structure are given: they concern the properties of infinite divisibility for the sup, and domains of attraction for Boolean functions. The last sections are devoted to the study of two particular Boolean functions; emphasis is put on the stereological implications of the approach. A critical example illustrates the theory.