Outer Measure and the Construction of Lebesgue Measure
Cover a set by intervals, take the infimum of cover lengths, and obtain an outer measure for every subset. The Caratheodory condition selects a sigma-algebra on which outer measure becomes countably additive.
What a learner can do afterwards
- Compute the outer measure of a countable set and of an interval
- State the Caratheodory condition and use it to test a set for measurability
- Describe how a non-measurable set is built and which axiom the construction relies on
1 · Read
Cover a set with countably many open intervals and add their lengths. The infimum of these totals is its outer measure. The infimum need not be attained by one cover. Every subset of the line gets an outer measure. A countable set has outer measure zero because its points can be covered by intervals with total length below every positive epsilon. An interval has its usual length.
Outer measure is monotone and countably subadditive. If E is contained in F, every cover of F also covers E, so the outer measure of E is at most that of F. Covers of countably many sets can be joined into a cover of their union, which gives an inequality rather than equality.
E is measurable when, for every test set A, the outer measure of A equals that of A intersect E plus that of A minus E. Subadditivity already bounds the outer measure of A by the sum of the parts, so only the reverse needs checking. A set of outer measure zero passes: the inside part has outer measure 0, and the outside part is at most the outer measure of A. The Caratheodory theorem says the passing sets form a sigma-algebra on which outer measure is countably additive.
A Vitali set starts with the relation x equivalent to y when x minus y is rational, usually on [0, 1]. Choose one representative from each equivalence class. Choosing from all classes uses the axiom of choice. The resulting set cannot receive a translation-invariant countably additive length extending interval length.
Outer measure begins with interval covers. The Caratheodory test selects a sigma-algebra where that outer measure becomes countably additive.
Lesson, questions and flashcards are original text written for this dataset.
Reading notes and questions are original text written for this dataset, and flashcards were drafted from collected fact candidates.
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