Regularity theorem for Lebesgue measure
Encyclopedia
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
".
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
).
B and a null set
N such that A is the symmetric difference
of B and N:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the regularity theorem for Lebesgue measure is a result in measure theory that states that Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
is a regular measure
Regular measure
In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...
. Informally speaking, this means that every Lebesgue-measurable subset of the real line is "approximately open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
" and "approximately closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
".
Statement 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
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
(i.e. – by the Heine-Borel theorem – closed and bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
).
Corollary: the structure of Lebesgue measurable sets
If A is a Lebesgue measurable subset of R, then there exists a Borel setBorel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
B and a null set
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
N such that A is the symmetric difference
Symmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
of B and N: