Let (S, S, P) be a finite measure space. A process $$ Y = {Y(f) $mid$ f : nonnegative measurable function on S} $$</TEX> is said to be a Poisson process with parameter $lambda$, where $lambda = P(S)$ if it has independent increments, in the sense that $Y(A_1), Y(A_2), ..., Y(A_k)$ are independent whenever $A_1, A_2,...,A_k$ are disjoint subsets of S, and the marginal distributions are Poisson with parameters $P(A_i)$. We can represent such a Poisson process as follows : Let ${U_i}$ be a sequence of independent identically distributed S-valued random variable and $N = Y(S)$. Then, for a function f on S, we can write $$ (1.1) Y(f) = sum^{N}_{i=1}{f(U_i)}. $$