Measure space §

From (2) and (3), a sigma-algebra is also closed under countable intersections and set differences. The smallest sigma-algebra containing a family $\mathcal{A}$ of subsets is denoted $\sigma(\mathcal{A})$ and is called the sigma-algebra generated by $\mathcal{A}$.

Example. On $X = \{1, 2, 3\}$, the sigma-algebra $\mathcal{F} = \{\varnothing, \{1\}, \{2,3\}, X\}$ is valid: it contains $X$, is closed under complement, and is closed under countable unions (here, finite unions). The Borel sigma-algebra $\mathcal{B}(\mathbb{R})$ is the smallest sigma-algebra containing all open subsets of $\mathbb{R}$. It is generated by the open intervals and contains all closed sets, $F_\sigma$ sets, $G_\delta$ sets, and more.

On $\mathbb{R}$, the Borel sigma-algebra contains:

• All open sets (by definition)
• All closed sets (complements of open sets)
• All countable sets (singletons are closed)
• All $F_\sigma$ (countable unions of closed) and $G_\delta$ (countable intersections of open) sets
• The Cantor set and its variants

It does not contain all subsets of $\mathbb{R}$: there are $2^{2^{\aleph_0}}$ subsets but only $2^{\aleph_0}$ Borel sets.

Borel hierarchy. The Borel sets are organised in a hierarchy built by transfinite alternation of countable unions and intersections:

• $\Sigma^1_0$: open sets
• $\Pi^1_0$: closed sets
• $\Sigma^1_1$: $F_\sigma$ sets (countable unions of closed)
• $\Pi^1_1$: $G_\delta$ sets (countable intersections of open)
• $\Sigma^1_2$, $\Pi^1_2$, … continuing up to level $\omega_1$

The Borel sigma-algebra is the union of all levels. Every continuous function $f: X \to \mathbb{R}$ is Borel-measurable because $f^{-1}(U)$ is open for every open $U$, and the preimage criterion (see Measurable functions) then applies.

Relation to Lebesgue measure. The standard construction of Lebesgue measure proceeds via outer measure: one defines an outer measure $\lambda^*$ on $\mathcal{P}(\mathbb{R})$, and the Carathéodory measurable sets (see below) form the Lebesgue sigma-algebra $\mathcal{L}$, which contains $\mathcal{B}(\mathbb{R})$. The passage from Borel to Lebesgue measurable sets is a completion (adding all subsets of null sets), not a Carathéodory extension.

Theorem (Carathéodory, 1914). If $\mu^*$ is an outer measure on $X$, then the collection of $\mu^*$-measurable sets forms a sigma-algebra, and $\mu^*$ restricted to this sigma-algebra is a countably additive measure.

Why the measurable sets form a sigma-algebra. The key observation is that the splitting condition $\mu^*(E) = \mu^*(E \cap A) + \mu^*(E \setminus A)$ is equivalent to $\mu^*(E) \leq \mu^*(E \cap A) + \mu^*(E \setminus A)$ (the reverse inequality follows from subadditivity). One verifies:

• $\varnothing$ and $X$ are measurable (trivial splitting)
• If $A$ is measurable, so is $X \setminus A$ (symmetry of the condition)
• If $A_1, A_2$ are measurable, so is $A_1 \cup A_2$ (using subadditivity and the splitting property iteratively)
• Countable unions follow by induction and monotonicity

Construction of Lebesgue measure.

1. Define the outer measure $\lambda^*$ on $\mathcal{P}(\mathbb{R})$ (see Measures for the formula).
2. The Carathéodory measurable sets form the Lebesgue sigma-algebra $\mathcal{L}$.
3. $\lambda = \lambda^*|_{\mathcal{L}}$ is Lebesgue measure.

Key properties:

• $\mathcal{B}(\mathbb{R}) \subseteq \mathcal{L}$ (every Borel set is Lebesgue measurable)
• $\mathcal{L}$ is strictly larger than $\mathcal{B}(\mathbb{R})$ (e.g., the Cantor set has measure zero but contains non-Borel subsets that are still Lebesgue measurable)
• Every subset of a null set is Lebesgue measurable (completeness)

Common examples:

A measure space is complete if every subset of a null set is measurable. Lebesgue measure on $\mathbb{R}$ is complete; Borel measure is not.

A measure space is $\sigma$-finite if $X = \bigcup_n A_n$ with $\mu(A_n) < \infty$. This is important for the Radon–Nikodym theorem and the uniqueness of extensions.

The collection of null sets is a sigma-ideal: it is closed under countable unions and under taking subsets. This means:

• If $N_1, N_2, \ldots$ are null sets, so is $\bigcup_n N_n$.
• If $N$ is null and $A \subseteq N$ with $A \in \mathcal{F}$, then $A$ is null.

This vocabulary is used throughout the rest of measure theory:

• In Lp spaces, functions are identified up to a.e. equality (equivalence classes mod null sets).
• In the Lebesgue integral, the dominated convergence theorem requires a.e. convergence, not pointwise convergence.
• The Radon–Nikodym theorem (in Measures) states that the density is unique only a.e.

Step 1: Verify sigma-algebra axioms.

• $X \in \mathcal{F}$ ✓
• Complements: $X \setminus \{1\} = \{2,3,4\} \in \mathcal{F}$ ✓; $X \setminus \{2,3\} = \{1,4\} \in \mathcal{F}$ ✓; $X \setminus \{4\} = \{1,2,3\} \in \mathcal{F}$ ✓; $X \setminus \{1,4\} = \{2,3\} \in \mathcal{F}$ ✓; $X \setminus \{2,3,4\} = \{1\} \in \mathcal{F}$ ✓; $X \setminus \{1,2,3\} = \{4\} \in \mathcal{F}$ ✓
• Countable unions: since $X$ is finite, all unions are finite. Checking: $\{1\} \cup \{2,3\} = \{1,2,3\} \in \mathcal{F}$ ✓; $\{1\} \cup \{4\} = \{1,4\} \in \mathcal{F}$ ✓; $\{2,3\} \cup \{4\} = \{2,3,4\} \in \mathcal{F}$ ✓; $\{1\} \cup \{2,3\} \cup \{4\} = X \in \mathcal{F}$ ✓

Step 2: Define the measure.

The sets $\{1\}$, $\{2,3\}$, $\{4\}$ form a partition of $X$. Every set in $\mathcal{F}$ is a union of some of these atoms. Define:

• $\mu(\varnothing) = 0$
• $\mu(\{1\}) = 2$, $\mu(\{2,3\}) = 5$, $\mu(\{4\}) = 1$
• $\mu(\{1,4\}) = 2 + 1 = 3$
• $\mu(\{2,3,4\}) = 5 + 1 = 6$
• $\mu(\{1,2,3\}) = 2 + 5 = 7$
• $\mu(X) = 2 + 5 + 1 = 8$

Countable additivity holds because the atoms are disjoint and every measurable set is a union of atoms.