for each t ∈ [0 , 1) and the “stopped backward” filtration G = {F$_{t}$$_{∧}$$_{λ}$ : 1 ≥ t ≥ 0 } . Further, introduce the following Definition 1. An estimator ˜ π$_{0}$ ( λ ) of the proportion π$_{0}$ of true nulls with tuning parameter λ ∈ [0 , 1) is said to be regular if it is measurable with respect to (wrt) F$_{λ}$ . It is crucial to note that, if λ is a functional of the p-values such that the information contained in F$_{λ}$ is not sufficient to determine the value of ˜ π$_{0}$ ( λ ), then ˜ π$_{0}$ ( λ ) is not measurable wrt F$_{λ}$ . This can happen for dynamic Storey procedures considered in Liang and Nettleton (2012), where λ is adaptively determined from the data. When this happens, ˜ t$_{α}$ ( λ ) does not have to be a stopping time wrt to G , the optional stopping theorem can not be applied, and the martingale arguments to prove the conservativeness of (dynamic) Storey FDR procedure in Storey et al. (2004) and Liang and Nettleton (2012) are invalid. Here is out main result: Theorem 1. If ˜ π$_{0}$ ( λ ) is regular, then ˜ t$_{α}$ ( λ ) is a stopping time with respect to G . Further, if ˜ t$_{α}$ ( λ ) < 1 , then The proof of Theorem 1 is given in Appendix A. Note that ˜ t$_{α}$ ( λ ) < 1 excludes the mean- ingless case α = 1 as the targeted FDR level. Therefore, ˜ t$_{α}$ ( λ ) < 1 in Theorem 1 essentially is not an assumption. Theorem 1 shows that, regardless of whether the p-values are independent, whether they are continuously distributed, or the stochastic orders of their distributions wrt the standard uniform random variable, the rejection threshold of a Storey-type FDR procedure is a stopping time wrt to the stopped backward filtration, when the estimator of the proportion of true nulls is regular. It also shows that a Storey-type FDR estimator equals the pre-specified FDR level at the threshold of its corresponding FDR procedure, when the estimator of the proportion of true nulls is regular and the corresponding decision rule is statistically meaning- fully implemented. Equally importantly, Theorem 1 shows that the STP may not hold for the rejection threshold of a Storey-type procedure when ˜ π$_{0}$ ( λ ) is not regular, i.e., when adaptivity of the estimator ˜ π$_{0}$ ( λ ) to F$_{λ}$ is not ensured. We make four remarks on Theorem 1 and its proof: (i) the measurability of ˜ π$_{0}$ ( λ ) wrt to F$_{λ}$ is critical to the STP of ˜ t$_{α}$ ( λ ); (ii) the fact that (6), i.e., ˜ F DR$_{λ}$ ( ˜ t$_{α}$ ( λ ) ) = α for ˜ t$_{α}$ ( λ ) < 1, 4 ˜ F DR$_{λ}$ ( ˜ t$_{α}$ ( λ ) ) = α. (6)