Exercise 1.5 (Easy properties of the unsigned integral)

Let $f,g:\mathrm{\Omega }\to [0,+\infty ]$ be measurable functions.

1.

(Superadditivity) We have ${\int }_{\mathrm{\Omega }}(f+g)d\mu \ge {\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}fd\mu +{\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}gd\mu$ and for any $c\in [0,+\infty ]$ we have ${\int }_{\mathrm{\Omega }}\mathit{cf}d\mu =c{\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}fd\mu$.

2.

(Almost everywhere equivalence) If $\mu$-almost everywhere $f=g$, then ${\int }_{\mathrm{\Omega }}fd\mu ={\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}gd\mu$.

3.

(Vanishing) If ${\int }_{\mathrm{\Omega }}fd\mu =0$, then $f$ is zero $\mu$-almost everywhere.

4.

(Monotonicity) If $\mu$-almost everywhere $f\le g$, then ${\int }_{\mathrm{\Omega }}fd\mu \le {\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}gd\mu$.

5.

(Markov’s inequality) For any $\lambda \in (0,+\infty )$: $$\mu \left(\left\{x\in X:f(x)\ge \lambda \right\}\right)\le \frac{1}{\lambda }{\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}fd\mu .$$

6.

(Compatibility with the simple integral) If $f$ is simple, then $\mathrm{Simp}{\int }_{\mathrm{\Omega }}fd\mu ={\int \; <!--\; nolimits\; -->}_{\mathrm{\Omega }}fd\mu$.

7.

(Compatibility with measure) For any measurable set $E$, we have ${\int }_{\mathrm{\Omega }}{1}_{E}d\mu =\mu (E)$.