Let $f,g $ be non negative measurable function and let $B$ be a measurable set. If $\int\_{A} f\, d \mu = \int\_{A} g \, d \mu $ for every $ A \subset B$ then $ f= g $ $\mu$-a.e in $B$ (suppose every integral is finite). May someone give me a hint for this one? I tried arguing by contradiction but $f $ and $g$ may have the same integral even on a set of positive measure on which they are not equal..