In the paper, we investigate the following fundamental question. For a set KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK in KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK, when does there exist an equivalent probability measure KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK such that KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK is uniformly integrable in KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK. Specifically, let KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK be a convex bounded positive set in KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK. Kardaras (J Funct Anal 266:1913–1927, 2014) asked the following two questions: (1) If the relative KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK-topology is locally convex on KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK, does there exist KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK such that the KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK- and KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK-topologies agree on KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK? (2) If KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK is closed in the KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK-topology and there exists KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK such that the KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK- and KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK-topologies agree on KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK, does there exist KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK such that KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK is KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK-uniformly integrable? In the paper, we show that, no matter KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK satisfying these desired properties. We also investigate the peculiar effects of KL0(P)QKL1(Q)KL1(P)L0(P)KQ∼PL0(Q)L1(Q)KKL0(P)Q∼PL0(Q)L1(Q)KQ′∼PKQ′KQ∼PK being positive.