Inside a fixed bounded half-open ambient interval, assign length b−a to each half-open subinterval (a, b], then add lengths across finite disjoint unions. This defines a premeasure on an algebra of elementary sets. The extension problem asks whether those initial assignments determine a countably additive measure on the much larger sigma-algebra they generate. Agreement on simple sets is valuable only if both existence and uniqueness are justified.
Predict
Predict the distinction between an algebra and a sigma-algebra. An algebra is closed under finite unions and complements, but a countable union may leave it. A premeasure must be countably additive whenever a disjoint countable union happens to remain in the algebra. It is not required to assign values to a union outside its domain. This conditional domain statement makes the extension theorem meaningful rather than circular.
Experiment
Follow the construction in three stages.
Stage
Domain
Required property
Premeasure μ0
Algebra A
Countable additivity when union lies in A
Cover outer measure μ*
Every subset
Monotonicity and subadditivity
Restricted measure
Carathéodory sets
Countable additivity
For an elementary set E, check both inequalities needed for μ*(E)=μ0(E), rather than relying only on its own one-set cover.
Understand
Define μ*(T) as the infimum of Σμ0(A_j) over countable algebra covers. The outer-measure proof uses the same summable-error argument as interval coverings. For E in the algebra, its own cover gives μ*(E)≤μ0(E). Given any cover A_j of E, disjointify E∩A_j by subtracting earlier cover sets. The resulting algebra pieces B_j have disjoint union E, so μ0(E)=Σμ0(B_j)≤Σμ0(A_j). This proves agreement. To show E measurable, split every cover A_j into A_j∩E and A_j outside E; finite additivity preserves its total cost. Taking infima yields μ*(T∩E)+μ*(T outside E)≤μ*(T). Carathéodory’s theorem now supplies an extension on a sigma-algebra containing the original algebra.
Apply
Uniqueness uses the pi-lambda theorem. A pi-system is closed under finite intersections; a lambda-system contains X, complements, and countable disjoint unions. The smallest lambda-system containing a pi-system is a sigma-algebra: first fix a generator A and use the lambda axioms to show the class of B with A∩B in the smallest lambda-system contains that whole system; then repeat with arbitrary A from the system. This proves closure under intersections, and disjointification gives arbitrary countable unions. For two finite measures agreeing on a generating pi-system that contains X, their agreement class is a lambda-system, so they agree everywhere generated.
Continue
If the premeasure is sigma-finite, choose an increasing algebra cover E_n with finite measure. Restrict both proposed extensions to E_n and apply finite uniqueness; continuity from below then gives agreement on every generated measurable set. Without sigma-finiteness, uniqueness is not guaranteed by this theorem. The conclusion concerns the generated sigma-algebra; a completed extension can contain additional subsets of null sets. Keep those domains explicit when claiming two measures are the same. This construction provides the foundation for Lebesgue measure and later product measures.
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A length premeasure assigns disjoint intervals (0,2] and (3,7]. Find their union measure.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Two probability measures agree on all atoms of a four-atom algebra. How many unions must then have equal measure?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove the cover outer measure agrees with a premeasure μ0 on its algebra.
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Why is every algebra set Carathéodory measurable for the induced outer measure?
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
State the finiteness condition for uniqueness of extension.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove a lambda-system that is also a pi-system is a sigma-algebra.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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