Measure theory begins by deciding which sets are observable and how size is assigned to countable unions. Sigma-algebras and measures encode those two decisions.
Sigma-algebras
A sigma-algebra is the domain on which countable set operations are legitimate.
Definition. A sigma-algebra \(\mathcal A\) on \(X\) is a collection of subsets containing \(X\), closed under complements, and closed under countable unions. The pair \((X,\mathcal A)\) is a measurable space.
Proposition. Arbitrary intersections of sigma-algebras are sigma-algebras. Therefore every collection \(\mathcal C\) generates a smallest sigma-algebra \(\sigma(\mathcal C)\).
Proof. Membership in an intersection means membership in every sigma-algebra in the family. The closure properties hold in each member, hence hold in the intersection. Taking the intersection of all sigma-algebras containing \(\mathcal C\) gives \(\sigma(\mathcal C)\).
Example (generated information). If \(T:X\to Y\) and \((Y,\mathcal B)\) is measurable, then \[ \sigma(T)=\{T^{-1}(B):B\in\mathcal B\} \] is the smallest sigma-algebra on \(X\) making \(T\) measurable. For real \(T\), \[ \sigma(T)=\sigma\bigl(\{T\le q\}:q\in\mathbb Q\bigr), \] because rational half-lines generate \(\mathcal B(\mathbb R)\).
Measures and continuity
Countable additivity is often used through monotone continuity.
Definition. A measure is a map \(\mu:\mathcal A\to[0,\infty]\) with \(\mu(\varnothing)=0\) and \(\mu(\bigcup_n A_n)=\sum_n\mu(A_n)\) for pairwise disjoint measurable sets \(A_n\).
Theorem. If \(A_n\uparrow A\), then \(\mu(A_n)\uparrow\mu(A)\). If \(A_n\downarrow A\) and \(\mu(A_1)\lt \infty\), then \(\mu(A_n)\downarrow\mu(A)\).
Proof. For increasing sets, decompose \(A\) into disjoint increments \(A_1\) and \(A_n\setminus A_{n-1}\). Countable additivity gives the limit. For decreasing sets, apply the increasing result to \(A_1\setminus A_n\uparrow A_1\setminus A\), using finiteness of \(\mu(A_1)\) to subtract.
Proposition (basic inequalities). If \(A\subset B\), then \(\mu(A)\le\mu(B)\), and if \(\mu(A)<\infty\), \[ \mu(B\setminus A)=\mu(B)-\mu(A). \] For arbitrary measurable \(A_n\), \[ \mu\!\left(\bigcup_{n=1}^{\infty}A_n\right)\le\sum_{n=1}^{\infty}\mu(A_n). \]
Proof. The disjoint decomposition \(B=A\sqcup(B\setminus A)\) gives monotonicity and the difference formula. For subadditivity, disjointify by \[ B_1=A_1,\qquad B_n=A_n\setminus\bigcup_{k<n}A_k. \] Then \(\bigcup_nB_n=\bigcup_nA_n\) and \(B_n\subset A_n\), so countable additivity and monotonicity give the result.
Pi-lambda uniqueness
Measures are often determined by their values on a smaller generating class.
Theorem (uniqueness on a generating \(\pi\)-system). Let \(\mathcal P\) be a \(\pi\)-system containing \(X\). If two finite measures agree on \(\mathcal P\), then they agree on \(\sigma(\mathcal P)\). The same holds for sigma-finite measures if \(X\uparrow P_n\) for sets \(P_n\in\mathcal P\) on which both measures are finite.
Proof. In the finite case let \(\mathcal D=\{A:\mu(A)=\nu(A)\}\). Equality on \(X\) and finiteness show that \(A\in\mathcal D\) implies \(X\setminus A\in\mathcal D\), while countable additivity gives closure under disjoint countable unions. Thus \(\mathcal D\) is a Dynkin system containing \(\mathcal P\). Dynkin's \(\pi\)-\(\lambda\) theorem yields \(\sigma(\mathcal P)\subset\mathcal D\).
In the sigma-finite case apply the finite argument to the restrictions \(A\mapsto\mu(A\cap P_n)\) and \(A\mapsto\nu(A\cap P_n)\). Agreement on \(\mathcal P\) follows because \(\mathcal P\) is closed under intersections. Then let \(n\to\infty\) and use continuity from below.
Theorem (Dynkin's \(\pi\)-\(\lambda\) theorem). If a Dynkin system \(\mathcal D\) contains a \(\pi\)-system \(\mathcal P\), then \(\sigma(\mathcal P)\subset\mathcal D\).
Proof. Let \(\lambda(\mathcal P)\) be the smallest Dynkin system containing \(\mathcal P\). For fixed \(A\in\mathcal P\), the class \[ \mathcal D_A=\{B:A\cap B\in\lambda(\mathcal P)\} \] is a Dynkin system containing \(\mathcal P\), hence equals \(\lambda(\mathcal P)\). Repeating with fixed \(B\in\lambda(\mathcal P)\) shows that \(\lambda(\mathcal P)\) is closed under finite intersections. A Dynkin system closed under finite intersections is a sigma-algebra, so \(\lambda(\mathcal P)=\sigma(\mathcal P)\subset\mathcal D\).
Completion
Completeness adds all subsets of null sets.
Proposition. Every measure space has a completion \((X,\overline{\mathcal A},\overline\mu)\), where \(\overline{\mathcal A}\) contains all subsets of \(\mu\)-null measurable sets.
Proof. A convenient description is \[ \overline{\mathcal A} =\{E\subset X:\text{there are }A,B\in\mathcal A,\ A\subset E\subset B,\ \mu(B\setminus A)=0\}. \] Equivalently, \(E\triangle A\) is contained in a measurable null set. This class is closed under complements and countable unions. Define \(\overline\mu(E)=\mu(A)\). If \(A'\) is another measurable representative, then \(A\triangle A'\) is null, so \(\mu(A)=\mu(A')\); the value is well-defined. Replacing disjoint completed sets by measurable representatives and discarding null overlaps proves countable additivity. Every subset of a completed null set again belongs to \(\overline{\mathcal A}\), so the extension is complete.
Measurable maps
Measurability is the condition that inverse images preserve observable sets.
Definition. A map \(f:(X,\mathcal A)\to(Y,\mathcal B)\) is measurable if \(f^{-1}(B)\in\mathcal A\) for every \(B\in\mathcal B\).
Remark. If \(\mathcal C\) generates \(\mathcal B\), it suffices to check \(f^{-1}(C)\in\mathcal A\) for \(C\in\mathcal C\), because \[ \{B\subset Y:f^{-1}(B)\in\mathcal A\} \] is itself a sigma-algebra. Measurable maps are closed under composition since \((g\circ f)^{-1}(C)=f^{-1}(g^{-1}(C))\).