Integrable real and complex functions, and their integrals

Let $f:X\to \mathbb{R}$ be measurable. Its positive and negative parts are $f^{+}:=\max \{f,0\},f^{-}:=\max \{-f,0\};$ they are measurable by Closure properties of measurable functions used by the integral and satisfy $f=f^{+}-f^{-}$ and $\mathrm{\mid }f\mathrm{\mid }=f^{+}+f^{-}$.

The Lebesgue integral of a real measurable function is defined whenever at most one of $\int f^{+}d\mu$ and $\int f^{-}d\mu$ is $+\mathrm{\infty }$, in which case $\int fd\mu :=\int f^{+}d\mu -\int f^{-}d\mu \mathrm{.}$ The function is integrable when both integrals are finite, equivalently when $\int \mathrm{\mid }f\mathrm{\mid }d\mu <+\mathrm{\infty }$.

For a complex measurable function $h=u+iv$ with $u=\mathrm{Re}h,v=\mathrm{Im}h$ (Real and imaginary parts, complex conjugation, and modulus), define $h$ to be integrable when $\mathrm{\mid }h\mathrm{\mid }$ is integrable, and then define $\int hd\mu :=\int ud\mu +i\int vd\mu \mathrm{.}$

Closure properties of measurable functions used by the integral

Real and imaginary parts, complex conjugation, and modulus