To integrate a function one first measures sets, and on the real line no countably additive notion of length can be assigned to every subset at once. The resolution is to measure only a rich collection of sets, a sigma-algebra, and to build the measure on it by extension from a primitive notion of size. This post supplies the construction the Lebesgue integral takes for granted, the Caratheodory extension theorem and the Lebesgue measure it produces. It uses the limit theory of sequences and completeness [1].
#Sigma-algebras
A sigma-algebra on a set is a collection of subsets containing , closed under complement, and closed under countable union. Its members are the measurable sets.
Closure under complement and countable union gives closure under countable intersection by De Morgan, and under set difference. The intersection of any family of sigma-algebras is a sigma-algebra, so for any collection of sets there is a smallest sigma-algebra containing it, the one generated by . On a metric space the sigma-algebra generated by the open sets is the Borel sigma-algebra, the measurable sets one cannot avoid.
#Measures and continuity
A measure on is a function with that is countably additive, meaning for every sequence of pairwise disjoint measurable sets.
Countable additivity forces monotonicity, since gives , and it forces continuity along monotone limits.
If then . If and then .
For the increasing case write the union as the disjoint union of the differences and . Countable additivity and then the telescoping finite sums give
For the decreasing case apply the increasing case to , which rises to . Then , and the finiteness of lets the subtractions cancel, giving the claim.
#Measurable functions and limits
A function is measurable when for every , equivalently when the preimage of every Borel set is measurable. The same criterion defines measurability for an extended-real , since it forces and into . Measurability survives every limit operation, the fact the convergence theorems rest on, with suprema and limits read as -valued.
If are measurable then , , , and are measurable, and so is wherever it exists.
The supremum is measurable because is a countable union of measurable sets. The infimum follows from . Then and are measurable as countable suprema and infima of measurable functions. Where the limit exists it equals the limit superior, hence is measurable.
#The Caratheodory extension theorem
A measure is hard to define directly, because countable additivity must be checked against all disjoint decompositions. The standard route defines an approximate size on all sets and then carves out those on which it behaves. An outer measure is a function with that is monotone and countably subadditive, . A set is Caratheodory measurable when it splits every set additively,
Subadditivity makes automatic, so the content of Equation (2) is the reverse inequality.
The Caratheodory measurable sets form a sigma-algebra , and the restriction of to is a measure.
The empty set is measurable and the definition is symmetric in and its complement, so contains and is closed under complement. For finite unions, let and split an arbitrary first by and then the part outside by ,
The first two pieces cover , so subadditivity bounds their sum below by , while the third is . Hence , and . For disjoint , testing the split of by gives , which by induction extends to for disjoint measurable . Now let be a countable disjoint union and . Since and , so monotonicity bounds the second term below by ,
Letting and using subadditivity on the tail, , so all are equalities. The first equality shows , so is closed under countable disjoint unions and hence, with complements, under all countable unions. Taking in the chain gives , the countable additivity of on .
The theorem is an engine. Feed it any outer measure and it returns a sigma-algebra carrying a genuine measure, and the measure is complete, since a subset of an -null set splits every set trivially and so is measurable.
#Lebesgue measure
On the line, length is the primitive size. Define the Lebesgue outer measure of by covering with open intervals and taking the cheapest cover,
It is monotone because a cover of a larger set covers the smaller, and countably subadditive because the inequality is trivial when some , and otherwise the union of covers of the , with the -th cover sharpened to within of , covers at cost within of . So is an outer measure, and Theorem 5 produces the Lebesgue sigma-algebra and Lebesgue measure on it.
Every interval is Lebesgue measurable, so the Borel sigma-algebra is contained in the Lebesgue sigma-algebra, and assigns each interval its length. Lebesgue measure is translation invariant and is the unique Borel measure giving each interval its length.
To see a half-line is measurable, take any and a near-optimal interval cover; splitting each covering interval at replaces it by its part in and the open interval covering the rest, with chosen so the endpoint stays covered without raising total length by more than . Thus , and gives the defining inequality. Half-lines generate the Borel sigma-algebra, so every Borel set is measurable. For the cover gives . For the reverse, fix and any open cover of . The compact subinterval has a finite subcover by the Heine-Borel theorem, and any finite open cover of an interval of length has total length exceeding , by induction on the cover size starting from an interval that contains the left endpoint and extends past it. Hence , and taking the infimum over covers and then gives . A singleton has by the cover , so subadditivity and monotonicity give as well, and every bounded interval shares the common length. Translation invariance follows because translating an interval cover preserves its total length. Uniqueness holds because two Borel measures agreeing with length on the intervals agree on the algebra of finite interval unions, a generating system closed under intersection, so they agree on every Borel set contained in some , the sets of the generating system that have finite measure and increase to . For a general Borel the sets increase to with finite measure, so continuity from below (Proposition 3, increasing case) gives , agreement on all Borel sets. The exhaustion of by the finite-measure intervals is what carries finite-measure agreement to the whole sigma-algebra.
With Lebesgue measure in hand the Lebesgue integral is built by integrating simple functions and passing to limits, the program the measures and integration post carries out on the structure produced here. Every measure that later posts use, the probability measures of the probability foundations and the Wiener measure of Brownian motion, is a Caratheodory extension of a primitive size.