Introduction

Prove that if $A$ and $B$ are two sets in $\mathcal{A}$ with $A \subseteq B$, then $\mu (A) \leq \mu (B)$. This property is called monotonicity.

Prove that if there is a set $A$ in the collection $\mathcal{A}$ for which $\mu (A) < \infty$, then $\mu (\emptyset ) = 0$.

Let $\left\{ E_{k}\right\}_{k = 1}^{\infty }$ be a countable collection of sets in $\mathcal{A}$. Prove that $\mu \left(\bigcup_{k = 1}^{\infty } E_{k}\right) \leq \sum_{k = 1}^{\infty } \mu \left(E_{k}\right)$.

A set-function $c$ is defined on all subsets of $\mathbb {R}$ as follows: define $c(E)$ to be $\infty$ if $E$ has infinitely many members and $c(E)$ to be the number of members in $E$ if $E$ is finite, and define $c(\emptyset ) = 0$. Show that $c$ is a countably additive and translation invariant set-function. This set-function is called the counting measure.