formula

$0&\leq &\limsup_n \int (f_n^+-f^+)d\mu\\ &\leq & \limsup_n \int (f_n^+-f^+)d\mu+\liminf\int (f_n^- -f^-)d\mu\\ &\leq & \limsup\int \{(f_n^+-f^+)+(f_n^--f^-)\}d\mu\\ &\leq & \limsup\int |f_n|d\mu -\int |f|d\mu=0$