Journal of the
Korean Mathematical Society JKMS

Let $(Z_1, {\CF}_1)$ and $(Z_2, {\CF}_2)$ represent two Brownianmeasures with ${\CF}_i \subset L^2({\S}_i, \mu_i)$, $i=1,2$. Clearly we can define $Z_1 \times Z_2$ on the field generated by ${\CF}_1\times {\CF}_2$ in an obvious way. The main purpose of the paperis to study the problem of constructing a productfunction $Z_1\times Z_2$ on as large a sub-family of $L^2({\S}_1\times {\S}_2, \mu_1 \times \mu_2)$ as possible and to characterizethose index families ${\CF}$ thatcontaining ${\CF}_1 \times {\CF}_2$, for which a `regular'extension of $Z_1\times Z_2$ exists. The regularity on thesample paths to be considered here is uniform continuity with respect to the ${ L}^2$-metric. The method used to measure the size of indexfamilies is a metric entropy.

Keywords: Brownian processes, metric entropy, product random measures, probability bounds, quadratic forms

MSC numbers: Primary 60E15; secondary60G17