3.2 Preliminaries Lemma 3.2.2. Suppose that T$_{n}$ and U$_{n}$ are m -currents for n = 1 , 2 , . . . and T$_{n}$ → T and U$_{n}$ → U in flat norm (i.e.,$_{F}$ ( T$_{n}$ − T ) → 0 ) for some m -currents T and U . The following properties hold: (a) αT$_{n}$ + βU$_{n}$ → αT + βU for any constants α, β ∈ $_{R}$, (b) ∂T$_{n}$ → ∂T , Convergence in the flat norm is linear and commutes with the boundary operator as the following easy lemma shows. The currents X and S have compact support by the definition of the flat norm. Thus X and S are normal by definition. Lemma 3.2.1. If T is a normal m -current and X and S are m - and ( m + 1) -currents such that T = X + ∂S and F ( T ) = M( X ) + M( S ) (i.e., T = X + ∂S is a flat norm decomposition of T ), then X and S are normal currents. Proof. By the definition of normal current, we have M( T ) + M( ∂T ) < ∞ . Thus so M( X ) < ∞ and M( S ) < ∞ . Since T = X + ∂S , we obtain Lastly, Our goal is to investigate conditions under which the flat norm decomposition of an integral current can be taken to be integral as well. The corresponding statement for normal currents is true and useful in our development. M( X ) + M( S ) = F ( T ) ≤ M( T ) < ∞ M( ∂X ) = M( ∂ ( X + ∂S )) = M( ∂T ) < ∞ . M( ∂S ) ≤ M( ∂S − T ) + M( T ) = M( − X ) + M( T ) < ∞ . 58