Regularity theorem for Lebesgue measure
- Regularity theorem for Lebesgue measure
In mathematics, the regularity theorem for Lebesgue measure is a result in measure theory that states that Lebesgue measure on the real line is a regular measure. Informally speaking, this means that every Lebesgue-measurable subset of the real line is "approximately open" and "approximately closed".
tatement of the theorem
Lebesgue measure on the real line, R, is a regular measure. That is, for all Lebesgue-measurable subsets "A" of R, and "ε" > 0, there exist subsets "C" and "U" of R such that
* "C" is closed; and
* "U" is open; and
* "C" ⊆ "A" ⊆ "U"; and
* the Lebesgue measure of "U" "C" is strictly less than "ε".Moreover, if "A" has finite Lebesgue measure, then "C" can be chosen to be compact (i.e. — by the Heine-Borel theorem — closed and bounded).
Corollary: the structure of Lebesgue measurable sets
If "A" is a Lebesgue measurable subset of R, then there exists a Borel set "B" and a null set "N" such that "A" is the symmetric difference of "B" and "N":
:
ee also
* Radon measure
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