If two measures agree on a generating -system and the space is a countable union of members having finite common measure, then they agree on the generated -algebra.
Lebesgue measure is the unique translation-invariant -finite Borel measure on normalized by . Translation invariance follows by comparing translated measure with on half-open intervals and applying uniqueness.
An open subset of the circle can be covered by countably many closed subarcs with disjoint interiors. Grouping a fixed enumeration into finite batches gives compact sets covering every point of the open set. By taking each finite batch long enough, the remaining total measure can be forced below any prescribed positive sequence tending to zero. This controls the measure of subsequent batches without assuming that a prescribed null subset is itself a countable union of compact null sets.
The Lebesgue measure of a measurable set is the supremum of the measures of its compact subsets. In particular, an open set of finite positive measure contains a compact subset with at least half its measure. Together with a finite subcover, this turns a finite Wiener covering lemma estimate into an estimate for arbitrary open superlevel sets.
An uncountable subset of the real numbers whose intersection with every null set for Lebesgue measure is countable. Testing only Borel sets suffices because every Lebesgue-null set has a Borel null superset. No measurability of is assumed.
If is a Lebesgue measurable set and , some measurable subset has Lebesgue measure . The function has Lipschitz continuity with constant , starts at and ends at ; the intermediate value theorem proves the assertion. Applying the assertion successively to remaining portions of gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most . This is a concrete consequence of non-atomic measure structure, which fails for general measures containing an atom of a measure.
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