 Therefore, a Caccioppoli set has a characteristic function whose total variation is locally bounded.From the theory of functions of bounded variation it is known that this implies the existence of a vector-valued Radon measure formula_21 such that

As noted for the case of general functions of bounded variation, this vector measure formula_21 is the distributional or weak gradient of formula_10.The total variation measure associated with formula_21 is denoted by formula_26, i.e.