Lebesgue Measure Theory
Generalizing Size, Exterior Measures, and Measurable Functions
The Lebesgue Measure was introduced to generalize our intuition of sizes to subsets of \(\mathbb{R}^n\) (Euclidean Space). Our intuition of sizes naturally comes from known geometric shapes like circles, rectangles, and cubes. We "cover" these subsets with these shapes and calculate the size of the sets based on the shapes that cover them.
Theorem 1: Every open subset \(\mathbb{O}\) of \(\mathbb{R}\) can be written uniquely as a countable union of disjoint intervals.
Theorem 2: Every open subset \(\mathbb{O}\) of \(\mathbb{R}^d\) (\(d \ge 1\)) can be written as a countable union of almost disjoint closed cubes.
Note: "Almost disjoint" means the interiors are disjoint, though boundaries may overlap.
Size of Intervals and Basic Shapes
A closed rectangle \(R\) in \(\mathbb{R}^d\) is defined by the product of intervals: \(R = [a_1, b_1]\times[a_2, b_2]\times\dots\times[a_d, b_d]\). The size of each interval is \(b_j - a_j\). The volume \(\left| R \right|\) is the product of these lengths.
Lemma 1: If \(R \subset \bigcup_{j = 1}^N R_j\), then \(\left| R \right| \leq \sum_{j = 1}^N \left| R_j\right|\).
The Exterior Measure
The exterior measure assigns a size to any subset \(E\) of \(\mathbb{R}^d\) by covering it with shapes from the outside. Formally, the exterior measure \(m_{\ast}\) is defined as:
where \(Q_j\) represents closed cubes.
Main observations regarding exterior measure:
- Monotonicity: If \(E_1 \subset E_2\) then \(m_{\ast}(E_1) \leq m_{\ast}(E_2)\).
- Countable Sub-Additivity: If \(E = \bigcup_{j = 1}^{\infty} E_j\), then \(m_{\ast}(E) \leq \sum_{j = 1}^{\infty} m_{\ast}(E_j)\).
Counterintuitive Sub-Additivity
Because we approximate from the outside, a set like \(E = [0,3]\) can be written as \(\bigcup_{j = 1}^{\infty}(0, 3-\tfrac{1}{j})\). Even though the sum of the measures of these subsets might approach infinity in some contexts, the exterior measure of the union remains 3. This confirms that \(m_{\ast}(E) \leq \sum m_{\ast}(E_j)\), but equality is not guaranteed without further restrictions.
Additivity conditions:
- If \(E = \bigcup Q_j\) (almost disjoint cubes), then \(m_{\ast}(E) = \sum \left|Q_j\right|\).
- If \(d(E_1, E_2) > 0\), then \(m_{\ast}(E_1 \cup E_2) = m_{\ast}(E_1) + m_{\ast}(E_2)\).
Measurability of Sets
We restrict the exterior measure to a \(\sigma\)-algebra called Lebesgue Measurable Sets, where the measure becomes countably additive.
Definition: A set \(\mathbb{E}\) is measurable if \(\forall \epsilon > 0\), there exists an open set \(\mathbb{O}\) with \(\mathbb{E} \subset \mathbb{O}\) such that \(m_{\ast}(\mathbb{O} - \mathbb{E}) \leq \epsilon\).
Basic measurable sets include closed sets, open sets, and null sets (measure 0). Countable unions and intersections of these are also measurable, leading to:
- \(G_{\delta}\): Countable intersection of open sets.
- \(F_{\sigma}\): Countable union of closed sets.
Measurability of Functions
Measurable functions are the building blocks of modern integration theory.
Definition: A function \(f\) on measurable set \(E\) is measurable if \(\forall a \in \mathbb{R}\), the set \(\{x \in E: f(x) < a\}\) is measurable.
The characteristic function \(\chi_E(x)\) (which is 1 if \(x \in E\) and 0 otherwise) allows us to build simple functions:
Properties & Convergence
In Lebesgue Theory, we often use the term almost everywhere (a.e.), meaning a property holds except on a set of measure zero.
- Limits of sequences of measurable functions are measurable.
- If \(f = g\) a.e. and \(f\) is measurable, then \(g\) is measurable.
- Fundamental Result: Any measurable function \(f\) is the pointwise limit of a sequence of simple functions \(\{\phi_k\}\).
This overview omits highly pathological cases like the Vitali Set or Egorov's Theorem, but establishes the framework for \(L^p\) spaces and robust integration.