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If T is integral, then P can be taken to be integral as well. 

Proof. This is a slight modification of Federer’s theorems which do not state Equa- tions (3.10b) and (3.10c) separately but rather a combined bound M( P ) + M( ∂P ) ≤ M( T ) + M( ∂T ) + ρ . We show only the derivation of the separated bounds. 

In the normal current case [19, 4.2.24], these bounds follow from Federer’s proof. In particular, we have currents P$_{1}$ , P$_{2}$ and Y such that P = P$_{1}$ + Y and the following bounds hold: 

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The bounds in Equations (3.10b) and (3.10c) follow from the triangle inequality and Equations (3.11a) to (3.11c): 

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In the integral current case [19, 4.2.21], Federer applies the approximation theorem 4.2.20 to obtain P close to the pushforward of T under a Lipschitz diffeomorphism f . 

