There are at least two distinct notions of an intensity function related to the theory of point processes.
In some literature, the intensity of a point process is defined to be the quantity
(1)
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provided it exists. Here, denotes probability. In particular, it makes sense to talk about point processes having infinite intensity, though when finite, allows to be rewritten so that
(2)
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as where here, denotes little-O notation (Daley and Vere-Jones 2007).
Other authors define the function to be an intensity function of a point process provided that is a density of the intensity measure associated to relative to Lebesgue measure, i.e.,if for all Borel sets in ,
(3)
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where denotes Lebesgue measure (Pawlas 2008).