Cousin's theorem


In real analysis, a branch of mathematics, Cousin's theorem states that:
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. 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.
In modern terms, it is stated as:

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 is a strictly positive real-valued function, while a tagged partition of is a finite sequence
Given a gauge and a tagged partition of , we say is -fine if for all, we have, where denotes the open ball of radius centred at. Cousin's lemma is now stated as: