The lift zonoid is a convenient representation of an integrable measure by a convex set in a higher-dimensional space. It is known that, under appropriate conditions, a uniformly integrable sequence of measures converges weakly if and only if the corresponding sequence of lift zonoids converges in the Hausdorff metric. We provide a new proof of this essential result. Our proof technique allows us to eliminate the unnecessary conditions previously considered in the literature. As a by-product, we obtain a characterization of uniform integrability, and a simple sufficient condition for tightness, of a sequence of integrable measures in terms of their lift zonoids.

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关于度量升力环带收敛性的说明

升力 zonoid 是高维空间中凸集的可积测度的方便表示。众所周知，在适当的条件下，当且仅当相应的升力 zonoid 序列收敛于 Hausdorff 度量时，一个均匀可积的度量序列会弱收敛。我们为这一基本结果提供了新的证明。我们的证明技术使我们能够消除文献中先前考虑的不必要条件。作为副产品，我们获得了一系列可积度量的均匀可积性特征和紧密性的简单充分条件，就其升力环带而言。