Optional and predictable projections

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Problem 16.8 from Bass R.F. "Stochastic Processes" [Bas11]More problems in Chapter 16 "The general theory of processes"

Show that if $X$ is bounded and measurable with respect to the product $\sigma$ -field $\mathcal{B}[0, \infty ) \times \mathcal{F}_{\infty }$ , then ${ }^{p}\left({ }^{o} X\right)={ }^{p} X$ .

Problem 16.9 from Bass R.F. "Stochastic Processes" [Bas11]More problems in Chapter 16 "The general theory of processes"

Suppose $T$ is a totally inaccessible stopping time. Show that if $X=\; \mathbb {1}_{[T, T]}$ , then ${ }^{p} X=0$ .

Problem 16.11 from Bass R.F. "Stochastic Processes" [Bas11]More problems in Chapter 16 "The general theory of processes"

Show that $\mu_{o}$ defined in (16.11) is a measure.

References

[Bas11] Bass R.F. Stochastic Processes. Cambridge University Press, 2011.