Summary: Suppose that $X=(X\sb t,F\sb t, t\in R\sb +)$ is an optional reward process with $(F\sb t)$ satisfying usual conditions. We correct the proof of existence about Snell envelope in [{\it J.-F. Mertens}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 45-68 (1972; Zbl 0236.60033)] and the proof of an important lemma (Lemma 4.6) in [{\it M. E. Thompson}, ibid. 19, 302-318 (1971; Zbl 0208.439)] and give a proof of the existence about Snell envelope under certain conditions, i.e. $EZ\sp -\sb \infty<\infty$ and $Z$ is upper-semi-continuous on the right or there is a stopping rule $τ\geσ$ such that $EZ\sp -\sb τ<\infty$ for any stopping rule $σ$. At the same time, we prove a four-repeated limit theorem when $Z$ is continuous on the right. The character and the uniqueness of the optimal stopping time or optimal stopping rule are discussed.