Cousin's theorem

In real analysis, a branch of mathematics, Cousin's theorem states that:

If for every point of a closed region (in modern terms, "closed and bounded") there is a circle of finite radius (in modern term, a "neighborhood"), then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of a given set having its center in the subregion.[1]

This result was originally proved by Pierre Cousin, a student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of $\mathbb {R} ^{n}$ . However, Pierre Cousin did not receive any credit. Cousin's theorem was generally attributed to Henri Lebesgue as the Borel–Lebesgue theorem. Lebesgue was aware of this result in 1898, and proved it in his 1903 dissertation.[1]

In modern terms, it is stated as:

Let

{\mathcal {C}}

be a full cover of [a, b], that is, a collection of closed subintervals of [a, b] with the property that for every x∈[a, b], there exists a δ>0 so that

{\mathcal {C}}

contains all subintervals of [a, b] which contains x and length smaller than δ. Then there exists a partition {I₁, I₂,...,I_n} of non-overlapping intervals for [a, b], where I_i=[x_i-1, x_i]∈

{\mathcal {C}}

and a=x₀ < x₁ <...< x_n=b for all 1≤i≤n.

In Henstock–Kurzweil integration

Cousin's theorem is instrumental in the study of Henstock–Kurzweil integration, and in this context, it is known as Cousin's lemma or the fineness theorem.

A gauge on $[a,b]$ is a strictly positive real-valued function $\delta :[a,b]\to \mathbb {R} ^{+}$ , while a tagged partition of $[a,b]$ is a finite sequence

P=\langle a=x_{0}<t_{0}<x_{1}<t_{1}<\cdots <x_{\ell -1}<t_{\ell -1}<x_{\ell }=b\rangle

Given a gauge $\delta :[a,b]\to \mathbb {R} ^{+}$ and a tagged partition $P$ of $[a,b]$ , we say $P$ is $\delta$ -fine if for all $j<\ell$ , we have $(x_{j},x_{j+1})\subseteq B{\big (}t_{j},\delta (t_{j}){\big )}$ , where $B(x,r)$ denotes the open ball of radius $r$ centred at $x$ . Cousin's lemma is now stated as:

If $a<b\in \mathbb {R}$ , then every gauge

\delta :[a,b]\to \mathbb {R} ^{+}

has a $\delta$ -fine partition.[2]

Notes

Hildebrandt 1925, p. 29
Bartle 2001, p. 11

References

Hildebrandt, T. H. (1925). The Borel Theorem and its Generalizations In J. C. Abbott (Ed.), The Chauvenet Papers: A collection of Prize-Winning Expository Papers in Mathematics. Mathematical Association of America.
Raman, M. J. (1997). Understanding Compactness: A Historical Perspective, Master of Arts Thesis. University of California, Berkeley. arXiv:1006.4131.
Bartle, R. G. (2001). A Modern Theory of Integration, Graduate Studies in Mathematics 32, American Mathematical Society.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[h1-1] Hildebrandt 1925, p. 29

[b1-2] Bartle 2001, p. 11