THE GENERALIZED FERNIQUE'S THEOREM FOR ANALOGUE OF WIENER MEASURE SPACE

In 1970, Fernique proved that there is a positive real number $\alpha$ such that $\int_{\mathbb{B}}\exp\{\alpha{\parallel}x{\parallel}^{2}\}dP(x)$ is finite where ($\mathbb{B},\;P$) is an abstract Wiener measure space and ${\parallel}\;{\cdot}\;{\parallel}$ is a measurable norm on ($\mathbb{B},\;P$) in [2, 3]. In this article, we investigate the existence of the integral $\int_{c}\exp\{\alpha(sup_t{\mid}x(t){\mid})^p\}dm_{\varphi}(x)$ where ($\mathcal{C}$, $m_{\varphi}$) is the analogue of Wiener measure space and p and $\alpha$ are both positive real numbers.