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In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:〔See: * * * *〕 :Let ''g''(''x'') be a real-valued continuous function on the interval (), and let ''E'' be the set of ''x'' ∈ (a,''b'') such that ''g''(''y'') > ''g''(''x'') for some ''y'' with ''x'' < ''y'' < ''b''. :Then ''E'' is an open set, and can be written as a disjoint union of intervals :: :such that ''g''(''a''''k'') = ''g''(''b''''k''), except possibly if ''a''''k'' = ''a'' when ''g''(''b''''k'') ≥ ''g''(''a''''k''). The colorful name of the lemma comes from imagining the graph of the function ''g'' as a mountainous landscape, with the sun shining horizontally from the right. The set ''E'' consist of points that are in the shadow. ==Proof== The set ''E'' is open, so it is composed of a countable disjoint union of intervals (''a''''n'', ''b''''n''). The main step is to show that ''g''(''b''''n'') ≥ ''g''(''x'') for ''x'' in (''a''''n'', ''b''''n''). If not take ''x'' with ''g''(''b''''n'') < ''g''(''x''). Let ''A'' be the closed subset of () consisting of points ''y'' such that ''g''(''y'') ≥ ''g''(''x''). It contains ''x'' but not ''b''''n''. It has a largest element, ''z'' say. Since ''z'' lies in ''E'', there is a ''y'' with ''z'' < ''y'' < ''b'' and ''g''(''y'') > ''g''(''z''). Since ''b''''n'' ∉ ''E'', ''g''(''t'') ≤ ''g''(''b''''n'') if ''b''''n'' ≤ ''t'' ≤ ''b''. Since ''g''(''y'') > ''g''(''z'') ≥ ''g''(''x'') > ''g''(''b''''n''), ''y'' must lie in (''z'', ''b''''n''). That contradicts the maximality of ''z''. Hence ''g''(''b''''n'') ≥ ''g''(''a''''n''). If ''a''''n'' ≠ ''a'', the reverse inequality holds. In fact since ''a''''n'' ∉ ''E'', ''g''(''y'') ≤ ''g''(''a''''n'') if ''a''''n'' ≤ ''y'' ≤ ''b''. So ''g''(''b''''n'') ≤ ''g''(''a''''n''). Hence ''g''(''b''''n'') = ''g''(''a''''n''). If ''g''(''x'') = ''g''(''a''''n'') at an interior point, then ''g''(''y'') ≤ ''g''(''x'') for ''x'' < ''y'' < ''b'', contradicting ''x'' ∈ ''E''. 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「Rising sun lemma」の詳細全文を読む スポンサード リンク
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