关于概率空间的等价性
On the equivalence of probability spaces
摘要：
对于一般测度空间$(M,\mathscr F, σ)$的sigma-代数$\mathscr F$所索引的一类高斯过程$W$，给出了这类高斯过程的二次变差表示有效的充要条件，从而将$A\in\mathscr F$的$σ(A)$恢复为$W$在$A$上的二次变差.我们还提供了一个谐波分析表示这类一般的过程。我们将这两个结果：$(i)$计算广义伊藤积分和$(ii)$证明了一个明确的，和测量理论的等价公式，实现了两种方法之间的等价高斯过程，其中一个选择的样本空间是传统的路径空间，和其他地方，它是施瓦茨空间的回火分布。
英文摘要：
For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\mathscr F$ of a general measure space $(M,\mathscr F, σ)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $σ(A)$, for $A\in\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path-space, and the other where it is Schwartz' space of tempered distributions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
