  It is natural to ask whether "positivity" of white noise operators can be
discussed in some sense and characterized. To answer this question, we consider
the Gel'fand triple over the Complex Gaussian space $(\ce'_c,\m_c)$, i.e.
$\ce'_c=\ce'+i\ce'$ equipped with the product measure $\m_c=\m'\times\m'$ where
$\m'$ is the Gaussian measure on $\ce'$ with variance 1/2 (Section
\ref{sec:2-2}). Following AKK's Legendre transform technique, we have
$\cw_{u_1,u_2}\subset L^2(\ce'_c,\m_c)\subset [\cw]^{*}_{u_1,u_2}$ for
functions $u_1,u_2\in C_{+,1/2}$ satisfying (U0)(U2)(U3). Several examples for
$u_1, u_2$ are given in Section \ref{sec:2-3}. We remark that Ouerdiane
\cite{oue} studied a special case $u_1(r^2)=u_2(r^2)=\exp(k^{-1}r^k)$, where
$1\leq k\leq 2$. In Section \ref{sec:3}, the characterization theorem for
measures can be extended to the case of positive product Radon measures on
$\ce'\times \ce'$. In addition, the notion of pseudo-positive operators is
naturally introduced via kernel theorem and characterized by an integrability
condition. Lemma \ref{lem:3-2} plays crucial roles in Section \ref{sec:3}.
