A sigma-algebra is closed under complements and countable unions. Generated sigma-algebras make this structure from a smaller family of sets; Borel sets arise from open sets. The pi-lambda theorem is a practical way to extend identities from generators to a whole sigma-algebra. Use the examples to compare finite partitions with the Borel structure on the real line.

A measure assigns nonnegative size with countable additivity. Distinguish finite and sigma-finite spaces, and interpret almost-everywhere statements through null sets. Continuity from below follows by decomposing an increasing union into disjoint increments; continuity from above requires a finite starting measure. Compare counting measure and point masses in the examples.

An outer measure assigns size to arbitrary subsets through countable covers. Carathéodory measurability singles out sets that split every other set additively. The extension theorem turns a premeasure on an algebra into a measure; its uniqueness conclusion needs the stated hypotheses. The examples connect the abstract construction to interval covers.

Lebesgue measure is constructed from elementary volumes and an outer measure. Borel sets are measurable, and completing the measure includes every subset of a null set. Regularity lets measurable sets be approximated by open and compact sets under the appropriate hypotheses. The examples show how null sets and ordinary volume fit this construction.

A measurable function pulls measurable sets back to measurable sets. Testing a generating family often suffices, and suitable pointwise limits preserve measurability. Increasing simple approximations to a nonnegative measurable function prepare the definition of integration. Work through a concrete approximation in the examples.

The Lebesgue integral begins with simple functions and extends by monotone approximation. Monotone convergence handles increasing nonnegative sequences; Fatou gives an inequality, while dominated convergence gives equality under an integrable bound. The examples show why pointwise convergence alone is insufficient.

Product measures extend rectangular size to a product sigma-algebra. Tonelli applies to nonnegative functions, while Fubini applies under integrability assumptions. Their distinction matters when signs or infinite values occur. Use the examples to check hypotheses before interchanging the order of integration.

Convergence in measure controls the size of exceptional sets, whereas almost-everywhere convergence describes individual points outside a null set. Egorov and Lusin link measurable behaviour to uniform or continuous behaviour under their stated hypotheses. The examples help identify which implications fail without extra conditions.

A probability space has total measure one. Random variables are measurable functions, their laws are pushforward measures, and expectation is integration. Change of variables lets an expectation be computed using a law instead of the original sample space. Check this translation in the examples.

Independence factors joint event probabilities. Borel–Cantelli connects sums of event probabilities with infinitely many occurrences; its two directions require different assumptions. Zero-one laws describe events whose probabilities must be extremal under specified independence structures. Use the examples to test precisely where independence enters.

The Radon–Nikodym theorem expresses an absolutely continuous measure as integration against a density, with uniqueness up to almost-everywhere equality. Lebesgue decomposition separates absolutely continuous and singular parts. The examples prepare the density construction used in conditional expectation.

Conditional expectation is measurable with respect to the retained sigma-algebra and agrees with the original variable’s integrals on its events. Radon–Nikodym gives existence, and the tower property describes successive conditioning. Work through conditioning on a finite partition in the examples.