标题： 概率括号表示法：概率空间，条件期望和初等鞅 显示英文标题

标题： Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales

摘要： 在本文中，我们继续探讨我们在之前文章中提出的概率括号表示法（PBN）的一致性和实用性。在简要回顾带有量纲分析的PBN后，我们通过引入与随机变量（R.V）相关或与随机过程（S.P）相关的概率空间，来研究PBN中的概率空间。接下来，我们用PBN表达条件期望（CE）的一些重要性质及其证明。然后，我们基于随机变量序列或基于滤链在PBN中引入鞅。在此过程中，我们看到PBN可以用来研究一些概率问题，否则可能需要显式使用测度论。在适用的情况下，我们使用量纲分析来验证我们的公式，并用图表来可视化PBN中的概念。我们希望这项研究表明，PBN，受量子力学（QM）中狄拉克符号启发并加以适应，可能在概率建模中成为一个有用的工具，至少对于那些已经熟悉QM中狄拉克符号的人来说是如此。 显示英文摘要

摘要： In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles. After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN. In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory. Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN. We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.