Exercise 0.23 (Simple consequences of the measure-theoretic axioms)

Let $\left(\mathrm{\Omega },{\mathcal{ℱ}}_{\mathrm{\Omega }},\mu \right)$ be a measure space.

1.

(Monotonicity) If $E\subseteq F$ are measurable, then $\mu (E)\le \mu (F)$.

2.

(Subadditivity) If $E_1,E_2,\dots$ are measurable, then $\mu \left({\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{E}_n\right)\le {\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }\mu (E_n)$.

3.

(Continuity from below) If $E_1\subseteq E_2\subseteq \dots$ are measurable, then $\mu \left({\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{E}_n\right)=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mu (E_n)$.

4.

(Continuity from above) If $E_1\supseteq E_2\supseteq \dots$ are measurable and $\mu (E_1)$ is finite, then $\mu \left({\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{E}_n\right)=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mu (E_n)$.