Outer Measures on Measurable Spaces

# Outer Measures on Measurable Spaces

Definition: Let $(X, \mathcal P(X))$ be a measurable space. An Outer Measure on this space is a set function $\mu^* : \mathcal P(X) \to [0, \infty]$ with the following properties:1) $\mu^*(\emptyset) = 0$.2) If $A$ and $B$ are subsets of $X$ and $A \subseteq B$ then $\mu^*(A) \leq \mu^*(B)$.3) If $(A_n)_{n=1}^{\infty}$ is ANY countable sequence of subsets of $X$ then $\displaystyle{\mu^* \left ( \bigcup_{n=1}^{\infty} A_n \right ) \leq \sum_{n=1}^{\infty} \mu^* (A_n)}$. |

The only example of an outer measure that we have seen so far is the Lebesgue outer measure on the measurable space $(\mathbb{R}, \mathcal P(\mathbb{R})$, which recall, is defined for all subsets $E$ of $\mathbb{R}$ by:

(1)\begin{align} \quad m^*(E) = \inf \left \{ \sum_{n=1}^{\infty} l(I_n) : E \subseteq \bigcup_{n=1}^{\infty} I_n \: \mathrm{and} \: \{ I_n = (a_n, b_n) : n \in \mathbb{N} \} \right \} \end{align}

Definition: Let $(X, \mathcal P(X))$ be a measurable space. An outer measure $\mu^*$ on this space is said to be Finite if $\mu^*(X) < \infty$. $\mu^*$ is said to be $\sigma$-Finite if $X$ is a countable union of subsets of $X$ with finite outer measure. |