  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.
