Definition 11.1.4 (Measures).label Let $E\suf\R^{n}$ we want to defined measure of $E$ denoted $\mu(E)$ that agrees with the Euclidean volume when $E$ is regular enough but has some further properties

1. (1)

   $\mu:\cali{P}(X)\to [0,\infty]$

2. (2)

   $\mu(E)\geq 0$

3. (3)

   $E\suf F\implies \mu(E)\leq \mu(F)$

4. (4)

   Given $\curl{E_n }_{1}^{\infty}$ such that $E_{n}\cap E_{m}=\emp$ for $n\neq m$ then $\mu \parens{\dcups{\infty }{n=1}E_n}=\sums{\infty}{n=1}\mu(E_{n})$