Birth-and-growth and nucleation and growth will be used as synonyms in this paper.

Then, denoted by [X.sub.i] the spatial location of the j-th nucleus of a site-saturated nucleation process, and by [[THETA].sup.t]([X.sub.j]) the grain obtained as the evolution up to time t > 0 of the nucleus [X.sub.j], the transformed region [[THETA].sup.t] at time t > 0 is

Then, a space-time nucleation process can be modeled by a marked point process, identifying [T.sub.i] as the time of birth of the i-th nucleus, and^ as its spatial location in [R.sup.d].

In other words, if N is a site-saturated nucleation process, then [LAMBDA](A) represents the mean number of nuclei born in A [subset] [R.sup.d]; whereas if N = [([T.sub.j], [X.sub.j]).sub.j] is a time-dependent nucleation process, then [LAMBDA]([s, t] x A') is the mean number of nuclei born in A' during a time interval [s, t].

Let us notice that whenever [V.sub.V] is independent of x (e.g., under assumptions of homogeneous nucleation and growth), then [V.sub.V] is independent of A and [V.sub.V](t) = [V.sub.V](t) (Rios and Villa, 2009; Stoyan et al, 1995, p.

Definition 1 (Causal cone) If N is a nucleation process on a suitable space K, then the causal cone C(t, x) of a point x at time t is the subset of K in which at least one nucleation event has to take place in order to cover the point x at time t.

Since, in general, the nucleation is random in time and space, then the transformed region at any time t > 0 will be a random set (Stoyan et al., 1995) in [R.sup.d], that is a measurable map from a probability space to the space of closed subsets in [R.sup.d].

Of course, different kinds of nucleation and growth models gives rise to different kinds of processes [{[[THETA].sup.t]}.sub.t].

As mentioned in the Introduction, we shall consider the case in which all the nucleation takes place at t = 0 (namely, site-saturated case), and the case in which the nucleation takes place in time (namely, time-dependent case).

Site-saturated nucleation processes and space-time dependent nucleation processes can be modeled by point processes and marked point processes, respectively.

The measure [LAMBDA] on [R.sup.d] and on [R.sub.+] x [R.sup.d], respectively, defined as [LAMBDA](A) := E[N(A)} for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , respectively, is called intensity measure of N; in other words, [LAMBDA](A) represents the mean number of nuclei born in A [subset] [R.sup.d] of a site-saturated process, and the mean number of nuclei born in A' during a time interval [s, t], where A = [s, t] x A' [subset] [R.sub.+] x [R.sup.d], of a time-dependent nucleation. We intentionally use the same notation N for the site-saturated nucleation process and for the time-dependent one because the site saturated process may be seen as a particular case of the time-dependent one by assuming [T.sub.j] [equivalent to] 0 for any j.

In this paper we shall assume Poissonian nucleation, that is N will be a Poisson process (in [R.sup.d] in the site-saturated case, and in [R.sub.+] x [R.sup.d] in the time-dependent case).