Lebesgue Measure
A Lebesgue Measure is a measurable function for subsets of n-dimensional Euclidean space that ...
- Context:
- …
- Example(s)
- a Lebesgue Outer Measure,
- a length measure of a closed interval [a,b] of real numbers, i.e. |b-a|.
- a cartesian product of closed intervals [a, b] and [c, d] of real numbers, (b − a)(d − c) .
- …
- Counter-Example(s)
- A Borel Measure,
- A Haar Meaure,
- A Hausdorff Measure.
- See: Lebesgue Measurable Set, Volume Form, Euclidean Space, Lebesgue Integration.
References
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Lebesgue_measure Retrieved:2018-9-16.
- 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 n-dimensional volume, n-volume, or simply volume. [1] It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.
The Lebesgue measure is often denoted by dx, but this should not be confused with the distinct notion of a volume form.
- 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 n-dimensional volume, n-volume, or simply volume. [1] It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.
2004
- (Isaev, 2004) ⇒ Alexander Isaev. (2004). “Introduction to Mathematical Methods in Bioinformatics." Springer. ISBN:3540219730,
- QUOTE: Definition 8.94. A function [math]\displaystyle{ f:\mathbb{R}\to \mathbb{R} }[/math] is called Lebesgue measurable or simply measurable if for every [math]\displaystyle{ b \in \mathbb{R} }[/math] the set [math]\displaystyle{ A_b(f)=\{x\in \mathbb{R}: f(x) \leq b }[/math] is Lebesgue measurable (that is, belongs to [math]\displaystyle{ \mathcal{L}(\mathbb{R}) }[/math]).
It is not hard to show that [math]\displaystyle{ f }[/math] is Lebesgue measurable if and only if [math]\displaystyle{ f^{-1}(E) }[/math] is Lebesgue measurable for every Borel set [math]\displaystyle{ E }[/math] in [math]\displaystyle{ \mathbb{R} }[/math] (see Sect. 6.7). Clearly, measurable functions are analogues of random variables for the Lebesgue measure.
We will now discuss the integrability of measurable functions on Lebesgue measurable sets in [math]\displaystyle{ \mathbb{R} }[/math]. Let first [math]\displaystyle{ A\in \mathcal{L}(\mathbb{R}) }[/math] be a set of finite measure, [math]\displaystyle{ \mu(A)\lt \infty }[/math]. Then the definition of integrability is identical to that for random variables …
- QUOTE: Definition 8.94. A function [math]\displaystyle{ f:\mathbb{R}\to \mathbb{R} }[/math] is called Lebesgue measurable or simply measurable if for every [math]\displaystyle{ b \in \mathbb{R} }[/math] the set [math]\displaystyle{ A_b(f)=\{x\in \mathbb{R}: f(x) \leq b }[/math] is Lebesgue measurable (that is, belongs to [math]\displaystyle{ \mathcal{L}(\mathbb{R}) }[/math]).