We prove that the range of sequence of vector measures converging widely satisfies a weak lower semicontinuity property, that the convergence of the range implies the strict convergence (convergence of the total variation) and that the strict convergence implies the range convergence for strictly convex norms. In dimension 2 and for Euclidean spaces of any dimensions, we prove that the total variation of a vector measure is monotone with respect to the range.

中文翻译：

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向量测度的范围收敛单调性和质量的范围单调性

我们证明向量度量的序列范围广泛收敛满足弱下半连续性，范围的收敛意味着严格的收敛（总变化的收敛），并且严格的收敛意味着严格凸范数的范围收敛。在维2中以及对于任何维数的欧几里德空间，我们证明向量测度的总变化相对于范围是单调的。