17/10/24

Reference: Lecture notes 3.3, 3.4

Lecture 6 Caratheodory theorem I.pdf

How do we construct measures? In some simple cases, for example when considering discrete probability, we just specify a $\sigma$-algebra $\mathcal{A}$ and the value of $\mu$ on each set from $\mathcal{A}$. But in more complicated cases it is useful to specify $\mu$ on a smaller class of subsets. For example, recall how we constructed the Lebesgue measure: first, we defined it on intervals, then we extended it to the Lebesgue outer measure by covering an arbitrary subset by open intervals. The outer measure did not have the properties we wanted so we restricted it to the class of measurable sets. It turns out that this procedure works in a general case. In the next few lectures, we discuss the following procedure for defining a measure:

  1. Define $\mu$ on a special class of subsets $\mathcal{S}$
  2. Extend it to an outer measure $\mu^*$ defined on all subsets by covering them with elements of $\mathcal{S}$
  3. Define the $\sigma$-algebra of measurable sets $\mathcal{A}$ and restrict $\mu^*$ to a measure $\mu$ for elements of $\mathcal{A}$

The first step is somewhat involved so let us start with the second step, as it is directly analogous to earlier discussion of the Lebesgue outer measure.

Definition. Let X be a set. A function $\mu^*\colon P(X) \to [0,\infty]$ is an outer measure if

  1. (Empty set) $\mu^*(\emptyset) = 0$

  2. (Monotonicity) If $A\subset B$ then $\mu^(A) \leq \mu^(B)$

  3. (Countably subadditivity) For any $A_1,A_2,\ldots \in P(X)$ we have

    $\mu^(\bigcup_{n=1}^\infty A_n) \leq \sum_{n=1}^\infty \mu^(A_n)$

Rembember that we defined the Lebesgue outer measure by covering an arbitrary set with open intervals. This construction can be generalized as follows.

Proposition. Let $\mathcal{S} \subset P(X)$ be a set containing $\emptyset$ and $X$, and let $\rho \colon \mathcal{S} \to [0,\infty]$ be such that $\rho(\emptyset) = 0.$ The function $\mu^* \colon P(X) \to [0,\infty]$ defined by

$\mu^*(A) = \inf \{ \sum_{n=1}^\infty \rho(E_n) \ | \ E_n \in \mathcal{S} \text{ and } A \subset \bigcup_{n=1}^\infty E_n \}$

is an outer measure.

Proof. Observe that the definition makes sense because there $X \in \mathcal{S}$ so there exists at least one way to cover $A$ with elements of $\mathcal{S}$. The proof is exactly the same as for the Lebesgue outer measure in Lecture 2, as an exercise verify the details. ∎

Example. If $\mathcal{S}=\{\emptyset,X\}$, then $\mu^*$ takes the same value on all non-empty subsets of $X$.

Example. Let $\mathcal{S}=P(X)$. Given $\rho \colon \mathcal{S} \to [0,\infty]$, the induced outer measure is $\mu^*(A) = \sum_{x \in A} \rho(x)$. This is actually a measure, as discussed in Lecture 5.