Random compact set
Put another way, a random compact set is a measurable function such that is almost surely compact and
is a measurable function for every .
(The distribution of а random compact convex set is also given by the system of all inclusion probabilities )
For , the probability is obtained, which satisfies
Thus the covering function is given by
Of course, can also be interpreted as the mean of the indicator function :
The covering function takes values between and . The set of all with is called the support of . The set , of all with is called the kernel, the set of fixed points, or essential minimum . If , is а sequence of i.i.d. random compact sets, then almost surely
and converges almost surely to
- Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.
- Molchanov, I. (2005) The Theory of Random Sets. Springer, New York.
- Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York.