Sigma-algebras and measures

The Riemann integral handles continuous functions on intervals but behaves badly under limits: a pointwise limit of Riemann integrable functions (e.g. indicators of finite subsets of $\mathbb{Q}\cap[0, 1]$, increasing to $\mathbf{1}_{\mathbb{Q}}$) need not be Riemann integrable. Lebesgue's theory measures sets first, then integrates.

1. σ-algebras (Def. 1.1) and Borel sets (Def. 1.2).
2. Measures (Def. 1.3) with examples (Example 1.4) and basic properties including continuity (Prop. 1.5).
3. Null sets and almost everywhere (Def. 1.6).

A σ-algebra on a set $X$ is a collection $\mathcal{F}$ of subsets containing $X$, closed under complements and under countable unions (hence countable intersections). $(X, \mathcal{F})$ is a measurable space and members of $\mathcal{F}$ are measurable sets.

For any collection $\mathcal{C}$ of subsets, $\sigma(\mathcal{C})$, the intersection of all σ-algebras containing $\mathcal{C}$, is the smallest σ-algebra containing it. The Borel σ-algebra $\mathcal{B}(\mathbb{R})$ is generated by the open sets; it is also generated by the intervals $(a, b]$, or by the rays $(-\infty, a]$.

A measure on $(X, \mathcal{F})$ is $\mu: \mathcal{F}\to[0, \infty]$ with $\mu(\emptyset) = 0$ and countable additivity: $\mu\big(\bigsqcup A_n\big) = \sum\mu(A_n)$ for pairwise disjoint $A_n$. $(X, \mathcal{F}, \mu)$ is a measure space; a probability space if $\mu(X) = 1$.

• Counting measure on $\mathcal{P}(X)$: $\mu(A) = |A|$ (possibly $\infty$).
• Dirac measure $\delta_x(A) = 1$ if $x\in A$, else $0$.
• Discrete probability: $\mu(A) = \sum_{x\in A}p_x$ with $p_x\ge0$, $\sum p_x = 1$.
• Lebesgue measure on $\mathcal{B}(\mathbb{R})$ with $\lambda((a, b]) = b - a$ (constructed later: see Section 2).

For measurable sets: (a) monotonicity $A\subseteq B\Rightarrow\mu(A)\le\mu(B)$; (b) countable subadditivity $\mu(\bigcup A_n)\le\sum\mu(A_n)$; (c) continuity from below: $A_n\uparrow A$ ⇒ $\mu(A_n)\to\mu(A)$; (d) continuity from above: $A_n\downarrow A$ and $\mu(A_1) < \infty$ ⇒ $\mu(A_n)\to\mu(A)$.

$N$ is a null set if $N\subseteq A$ for some measurable $A$ with $\mu(A) = 0$. A property holds almost everywhere (a.e.) if it fails only on a null set. Countable unions of null sets are null. A measure space is complete if every subset of a measure-zero set is measurable.

• Check the domain is a σ-algebra (countable unions, complements).
• Check $\mu(\emptyset) = 0$ and countable additivity on disjoint sets.
• For limits of sets use continuity, remembering the finiteness condition from above.