Chapter 2 - Measure Notes

Source: Axler, Measure, Integration & Real Analysis (MIRA), Chapter 2.

Outer Measure

Let \(A \subseteq \mathbb{R}\). The outer measure of \(A\) is

\[ |A| := \inf \left\{ \sum_{k \in \mathbb{N}} \ell(I_k) \ \middle|\ \begin{array}{l} \bullet\ I_k \text{ is an open interval for all } k \in \mathbb{N}, \\ \bullet\ A \subseteq \bigcup_{k \in \mathbb{N}} I_k \end{array} \right\}. \]

\(\sigma\)-Algebra

Let \(X\) be a set and let \(\mathcal{S}\) be a set of subsets of \(X\). Then \(\mathcal{S}\) is a \(\sigma\)-algebra on \(X\) if:

(i) \(\varnothing \in \mathcal{S}\).

(ii) If \(E \in \mathcal{S}\), then \(E^c = X \setminus E \in \mathcal{S}\).

(iii) If \(\{E_k : k \in \mathbb{N}\} \subseteq \mathcal{S}\), then

\[ \bigcup_{k \in \mathbb{N}} E_k \in \mathcal{S}. \]

That is, \(\mathcal{S}\) contains the empty set, is closed under complements, and is closed under countable unions.

Measurable Space

A measurable space \((X, \mathcal{S})\) is a set \(X\) together with a \(\sigma\)-algebra \(\mathcal{S}\) on \(X\).

Borel Set

The collection of Borel subsets of \(\mathbb{R}\) is the smallest \(\sigma\)-algebra on \(\mathbb{R}\) that contains all open subsets of \(\mathbb{R}\).

A set \(E \subseteq \mathbb{R}\) is Borel if \(E\) is in this collection.

Measurable Function

Let \((X, \mathcal{S})\) be a measurable space. A function \(f : X \to \mathbb{R}\) is measurable if

\[ f^{-1}(B) \in \mathcal{S} \]

for every Borel set \(B \subseteq \mathbb{R}\).

Borel Measurable Function

Let \(X \subseteq \mathbb{R}\). A function \(f : X \to \mathbb{R}\) is Borel measurable if \(f^{-1}(B)\) is Borel for every Borel set \(B \subseteq \mathbb{R}\).

Measure

Let \((X, \mathcal{S})\) be a measurable space. A function \(\mu : \mathcal{S} \to [0, \infty]\) is a measure if:

(i) \(\mu(\varnothing) = 0\).

(ii) For every disjoint sequence \(\{E_k : k \in \mathbb{N}\} \subseteq \mathcal{S}\),

\[ \mu\left(\bigcup_{k \in \mathbb{N}} E_k\right) = \sum_{k \in \mathbb{N}} \mu(E_k). \]

Measure Space

A measure space \((X, \mathcal{S}, \mu)\) consists of a set \(X\), a \(\sigma\)-algebra \(\mathcal{S}\) on \(X\), and a measure \(\mu\) on \((X, \mathcal{S})\).

Lebesgue Measure

Lebesgue measure is outer measure restricted to the Borel measurable space:

\[ (\mathbb{R}, \mathcal{B}, \mu), \qquad \mu = |\cdot|. \]

Lebesgue Measurable Set

A set \(A \subseteq \mathbb{R}\) is Lebesgue measurable if there exists a Borel set \(B \in \mathcal{B}\) such that

\[ B \subseteq A \quad \text{and} \quad |A \setminus B| = 0. \]

Sets of Outer Measure Zero

If \(|A| = 0\), then

\[ |A \setminus \varnothing| = |A| = 0 \quad \text{and} \quad \varnothing \subseteq A. \]

Therefore \(A\) is Lebesgue measurable.

Thus the Lebesgue measurable sets are Borel sets unioned with some set with outer measure \(0\):

\[ A = B \cup (A \setminus B). \]

Thus it is the smallest \(\sigma\)-algebra containing the Borel sets and sets with measure \(0\); see Axler 2.72. Hence

\[ \mathcal{L} = \{A \subseteq \mathbb{R} : A \text{ is Lebesgue measurable}\} \]

is a \(\sigma\)-algebra on \(\mathbb{R}\).

Remark

Already have: countable implies outer measure zero, which implies Lebesgue measurable.

So if a set is not Lebesgue measurable, then it is not countable. We showed in Axler 2.81 that there exists a Lebesgue measurable set \(A \subseteq [0,1]\) such that \(|A| = 0\) and \(\Lambda(A)\) is not a Lebesgue measurable set. Hence \(\Lambda(A)\) is not countable.