I am reading the proof of the compactness of Integral varifolds on L.Simon's book " Lecture on geometric measure theory", there is a part of the proof concerning the conclusion that the density is integral ( page249) which is intriguing me, the problem is next :
Given a sequence of integral valued functions $\psi_i$ converging weakly to a constant function $\theta_0$ on the unit ball of $\mathbb{R^n}$, assume that there exist sets $A_i$ contained in the unit ball such that $$ \psi_i \leq \theta_0 + \epsilon_i \hspace{0.2cm} \forall x \in A_i^{c}, \hspace{0.4cm} \mathcal{L}^n (A_i) \rightarrow 0 , \hspace{0.2cm} \epsilon_i \rightarrow 0 $$
The author deduced that $\theta_0$ is an integer by noticing that $min\lbrace{N,\psi_i \rbrace}$ converge $L^1$ to $\theta_0$, $\forall N > \theta_0$ which is not clear for me !!!