# Range convergence monotonicity for vector measures and range monotonicity of the mass

## TL;DR

This paper investigates the properties of vector measures, establishing conditions under which their range convergence implies strict convergence and demonstrating monotonicity of total variation with respect to the range in Euclidean spaces.

## Contribution

It proves weak lower semicontinuity of the range for vector measures and establishes the monotonicity of total variation relative to the range in Euclidean spaces.

## Key findings

- Range convergence implies strict convergence under certain conditions.
- Total variation is monotone with respect to the range in Euclidean spaces.
- Weak lower semicontinuity of the range is established for converging vector measures.

## Abstract

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.