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For varifolds whose first variation is representable by integration, weintroduce the notion of indecomposability with respect to locally Lipschitzianreal valued functions. Unlike indecomposability, this weaker connectednessproperty is inherited by varifolds associated with solutions to geometricvariational problems phrased in terms of sets, $G$ chains, and immersions; yetit is strong enough for the subsequent deduction of substantial geometricconsequences therefrom. Our present study is based on several further conceptsfor varifolds put forward in this paper: real valued functions of generalisedbounded variation thereon, partitions thereof in general, partition thereofalong a real valued generalised weakly differentiable function in particular,and local finiteness of decompositions.

A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds