Let $W$ be a standard Brownian motion under given probability space. For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time $T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = W(t ) 1_{(t\le T^a)} + W(T^a) 1_{(t> T^a)}$\wedge T^a)$. We want to consider the following question: Is the process$W^1$of the class DL? (Solution1): Yes. Indeed, for any fixed$t>0$, we can prove the collection of random variables$( W(s), 0< s< t)$is uniformly integrable by definition, since$E [|W^1(t)|] < \infty$. We provide completely different answer using the following proposition from the Problem 1.5.19 (i) of Book [Karazas and Shereve 98]. [Proposition] A local martingale of class DL is martingale. (Solution2): No.$W^1$is strict local martingale, since$E [W^1(T^1)] = 1> E [W(0)]$. By [Proposition],$W^1$is not of class DL. In the above, we obtained completely two different solutions. Where is wrong? 2 added 28 characters in body Let$W$be a standard Brownian motion under given probability space. For a given constant$a$,$W^a$is a truncated Brownian motion by stopping time$T^a = \inf{t>0: W(t) inf(t>0:W(t) = a}$a)$. That is, $W^a(t) = W(t) 1_{{t\le T^a}1_{(t\le T^a)} + W(T^a) 1_{{t> T^a}}$1_{(t> T^a)}$. We want to consider the following question: Is the process$W^1$of the class DL? [Solution1]: (Solution1): Yes. Indeed, for any fixed$t>0$, we can prove the collection of random variables${W(s):0 ( W(s), 0< s< t)$is uniformly integrable by definition, since$E [|W^1(t)|] < \infty$. We provide completely different answer using the following proposition from the Problem 1.5.19 (i) of Book [Karazas and Shereve 98]. [Proposition] A local martingale of class DL is martingale. [Solution2]: (Solution2): No.$W^1$is strict local martingale, since$E [W^1(T^1)] = 1> E [W(0)]$. By [Proposition],$W^1\$ is not of class DL.