On Relations Between Strict-Sense and Wide-Sense Conditional Expectations
On Relations Between Strict-Sense and Wide-Sense Conditional Expectations
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论严格意义上的条件期望与广义条件期望之间的关系
DOI：
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发表时间：
1957
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通讯作者：
Z. Šidák
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作者：
Z. Šidák
在本文中，我们研究了严格意义上的条件期望之间的关系（即普通的条件期望）和广泛的意义（即希尔伯特空间的预测$ l_2 $ $ l_2 $ in DOOB [3 ] .let $（x，f，mu）$是概率空间，$ g子集f $是$ sigma $ -Algebra。我们用$ l_2（g）$表示所有$ l_2 $的所有G-测量随机变量的系统。在巴哈杜尔之后，我们命名这样的子空间，$ l_2（g）$，可测量的子空间以及对它们的可测量预测的投影。本文最重要的结果是以下理论3.系统$ Mathfrak {M Mathfrak {M}子集L_2 $是一个可测量的子空间如果并且仅限于我，则满足以下条件：（0）$ mathfrak {m} $是封闭的线性子空间，（1）$ 1 Mathfrak {m} $，（2）如果在Mathfrak {m} $中$ f，Mathfrak {m} $中的$ g，则在Mathfrak {m} $中$ max（f，g）。使用了两个函数的操作，但仅用于边界...
In this paper we investigate the relations between conditional expectations in the strict sense (i.e. ordinary conditional expectations) and in the wide sense (i.e. projections of a Hilbert space $L_2 $ into some closed linear subspace), as they were defined by Doob [3].Let $(X,F,mu )$ be a probability space and $G subset F$ be some $sigma $-algebra. We denote by $L_2 (G)$ the system of all G-measurable random variables of $L_2 $. Following Bahadur we name such subspaces, $L_2 (G)$, measurable subspaces, and projections on them measurable projections.The most important result of the paper is the followingTheorem 3.The system$mathfrak{M} subset L_2 $is a measurable subspace if and only i f it satisfies the following conditions: (0) $mathfrak{M}$ is a closed linear subspace, (1) $1 in mathfrak{M}$, (2) if$f in mathfrak{M}$, $g in mathfrak{M}$, then$max (f,g) in mathfrak{M}$.In papers published up to now the operation of multiplying two functions was used, but it was introduced only for bounde...