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flat norm of an integral chain in codimension 1 has an optimal integral current decom- position; by the compactness theorem from geometric measure theory, the limit of these decompositions is also integral. 

In order to show that an in tegral current T has integral flat norm decomposition, we therefore find suitable simplicial approximations to T and take the limit of their simplicial flat norm decomp ositions to obtain an integral decomposition for T . 

We must also show that this decomposition achieves the flat norm value for T (that is, express T using integral curren ts in such a way that it remains an optimal flat norm decomposition). This is immediate if our simplicial approximations to T have simplicial flat norm values that converge to the flat norm of T but this is not necessary (see 

