Exercise 1.9 (Basic properties of the complex integral)

Let $f,g:\mathrm{\Omega }\to C$ be real-valued absolutely integrable functions, and let $c\in C$.

1.

(Linearity) Show that $f+g$ and $\mathit{cf}$ are also complex-valued absolutely integrable functions, with $${\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}f+g\mathit{d\mu }={\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}f\mathit{d\mu }+{\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}g\mathit{d\mu }$$

and

$${\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}\mathit{cf}\mathit{d\mu }=c{\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}f\mathit{d\mu }.$$

(For the second relation, one may wish to first treat the special cases $c>0$ and $c=-1$.)

2.

(Equivalence) Show that if $f$ and $g$ are equal almost everywhere, then $${\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}f\mathit{d\mu }={\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}g\mathit{d\mu }.$$

3.

Show that ${\int }_{\mathrm{\Omega }}\left|f\right|\mathit{d\mu }\ge 0$, with equality if and only if $f$ is zero almost everywhere.

4.

(Markov inequality) Show that $\mu (\{\omega :|f(\omega )|\ge t\})\le \frac{1}{t}{\int }_{\mathrm{\Omega }}\left|f\right|\mathit{d\mu }$ for any $0<t<\infty$.