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For a finite signed or complex measure $\mu$, its variation is the positive measureIt controls integration by $|\int fd\mu |\le \int |f|d|\mu |$.

A finite-dimensional vector measure has scalar measures as components and a total variation norm of a measure defined using their joint vector norm. Finite variation controls every pairing with a bounded continuous vector test. Derivatives in the bounded-variation space are examples.

A vector Radon measure has Radon measures as components. The vector form of the Riesz-Markov-Kakutani representation theorem identifies finite such measures with bounded functionals on $C_0(\Omega ;{\mathbb{R}}^n)$.

Convergence in the displayed sense combines strong function convergence with weak-star convergence of derivative vector measures against $C_0$ tests. Strong $L^1$ convergence and a uniform variation bound imply the derivative convergence by integration by parts and uniform test-function approximation. Conversely the Uniform boundedness principle bounds the derivative measures of a convergent sequence.

This is the coarsest topology making all displayed bounded-continuous test integrals continuous. On a separable metric space it is metrized by the bounded-Lipschitz metric. This probability-measure topology should be distinguished from the weak topology of a Banach space.