If $f$ and $g$ are nonnegative Lebesgue measurable functions, then we know that $\int (f+g) d\lambda = \int f d \lambda + \int g d \lambda $. Given the difinition of integral of an arbitrary Lebesgue measurable function, that is $ \int f d\lambda = \int f^+ d\lambda - \int f^- d\lambda$, how do you prove $\int (f_1 - f_2) d\lambda = \int f_1 d\lambda - \int f_2 d\lambda $ for nonnegative Lebesgue integrable functions $f_1$ and $f_2$ ?
That $f_1 - f_2$ is Lebesgue integrable is easily seen. Now, I can show that $\int (-f) d\lambda = -\int f d\lambda$, as $\int (-f) d\lambda = \int (-f)^+ d\lambda - \int (-f)^- d\lambda = \int f^- d\lambda - \int f^+ d\lambda = -\int fd\lambda$. Any ideas how to proceed?