Random Compact Set - Discussion

Discussion

Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently their distribution is given by the probabilities

for

(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

for

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