|
In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard Helly. The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures. ==Statement of the theorem== Let ''U'' be an open subset of the real line and let ''f''''n'' : ''U'' → R, ''n'' ∈ N, be a sequence of functions. Suppose that * (''f''''n'') has uniformly bounded total variation on any ''W'' that is compactly embedded in ''U''. That is, for all sets ''W'' ⊆ ''U'' with compact closure ''W̄'' ⊆ ''U'', :: :where the derivative is taken in the sense of tempered distributions; * and (''f''''n'') is uniformly bounded at a point. That is, for some ''t'' ∈ ''U'', ⊆ R is a bounded set. Then there exists a subsequence ''f''''n''''k'', ''k'' ∈ N, of ''f''''n'' and a function ''f'' : ''U'' → R, locally of bounded variation, such that * ''f''''n''''k'' converges to ''f'' pointwise; * and ''f''''n''''k'' converges to ''f'' locally in ''L''1 (see locally integrable function), i.e., for all ''W'' compactly embedded in ''U'', :: * and, for ''W'' compactly embedded in ''U'', :: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Helly's selection theorem」の詳細全文を読む スポンサード リンク
|