$f$ is Lebesgue integrable over $A$, and $B$ is a measurable subset of $A$. I want to show $\int_B f=\int_Af\chi_B$, where $\chi_B$ is the characteristic function of $B$ (it is 1 on B and 0 otherwise). My definition of integrable is "$f$ is integrable if the functions $f^+, f^-$ are integrable."
What I have tried to do is to write $\int_A f\chi_B=\int_Af\chi_B+\int_{A-B}f\chi_B,$ and it would simply follow. However, it is not obvious that I can write that since it is not given that $f$ is bounded. A hint would be helpful.