  Generalized conditional expectations, optional projections and predictable
projections of stochastic processes play important roles in the general theory
of stochastic processes, semimartingale theory and stochastic calculus. They
share some important properties with ordinary conditional expectations. While
the characterization of ordinary conditional expectations has been studied by
several authors, no similar work seems to have been done for these three
concepts. This paper aims at undertaking this task by giving
And$\hat{o}$-Douglas type characterization theorem for each of them.
