{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:34654eec-cd5e-4045-aa32-cbbe8fd63052", "labels": { "zh-CN": "法图引理", "en": "Fatou’s lemma" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "6b605f143a9c0c5dc4b3c5c0a16ff2552fc960a8020054020d4156c33726cb75", "occurrences": [ { "concept_id": "urn:uuid:34654eec-cd5e-4045-aa32-cbbe8fd63052", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-11/03-lebesgue-integration", "anchor": "first-2783e36cf95c02d27031-zh", "chapter": "11", "section": "11.3", "offset": 11089, "range": [ 16156, 17439 ], "source_revision": "dfa6fe6e07ccffde2e8fb24fc589de231194b6bc2966051e34615cb4e3844381", "source_category": "translation", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "zh/chapter-11/03-lebesgue-integration", "lang": "zh-CN", "section": "11.3", "signal": "named-theorem", "label": "法图引理", "english": "Fatou’s lemma", "offset": 10042 } }, "label": "> [!theorem] 引理 11.10 · 法图引理 (Fatou’s lemma)", "context": { "range": [ 16156, 17439 ], "quote": "\n\n> [!theorem] 引理 11.10 · 法图引理 (Fatou’s lemma)\n>\n> （法图）若 $(f_n)$ 是定义在测度空间 $( X , { \\mathcal { M } } , \\mu )$ 上的任意非负可测函数序列，则\n>\n> $$\n> \\int \\liminf _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu \\leq \\liminf _ {n \\to \\infty} \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n\n> [!proof]- 证明\n>\n> 函数 $\\begin{array} { r } { G _ { n } ( x ) = \\operatorname* { i n f } _ { k \\geq n } f _ { k } ( x ) } \\end{array}$ 构成一个非负可测函数的递增序列，使得对于所有 $n$ 都有 $G _ { n } \\leq f _ { n }$。因此，序列 $( f _ { n } )$ 的下极限是序列 $G _ { n }$ 的极限。\n>\n> $$\n> \\liminf _ {n \\to \\infty} f _ {n} = \\sup _ {n} G _ {n} = \\lim _ {n \\to \\infty} G _ {n}.\n> $$\n>\n> 由单调收敛定理 11.7，\n>\n> $$\n> \\lim _ {n \\to \\infty} \\int G _ {n} \\mathrm{d} \\mu = \\int \\operatorname * {l i m i n f} _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu\n> $$\n>\n> 而不等式 $G _ { n } \\leq f _ { n }$ 则意味着\n>\n> $$\n> \\int G _ {n} \\mathrm{d} \\mu \\leq \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n>\n> 因此\n>\n> $$\n> \\int \\operatorname * {l i m i n f} _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu = \\lim _ {n \\to \\infty} \\int G _ {n} \\mathrm{d} \\mu \\leq \\operatorname * {l i m i n f} _ {n \\to \\infty} \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n", "quote_sha256": "d6053d6a5361fe70b8a612e447e64dbbd4640080e94abf31f08c925dd0cd572e", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-11/03-lebesgue-integration", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-a7b4c1cac1e14d31671e12db" }, { "concept_id": "urn:uuid:34654eec-cd5e-4045-aa32-cbbe8fd63052", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-11/03-lebesgue-integration", "anchor": "first-2783e36cf95c02d27031-en", "chapter": "11", "section": "11.3", "offset": 13509, "range": [ 18104, 19505 ], "source_revision": "b10b1ac3b768247d93cf10ab40a926dfa4a8d3902771e5fca99cf3d02e26bc63", "source_category": "original_transcription", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "en/chapter-11/03-lebesgue-integration", "lang": "en", "section": "11.3", "signal": "named-theorem", "label": "Fatou’s lemma", "english": "Fatou’s lemma", "offset": 13226 } }, "label": "> [!theorem] Lemma 11.10 · Fatou’s lemma", "context": { "range": [ 18104, 19505 ], "quote": "\n\n> [!theorem] Lemma 11.10 · Fatou’s lemma\n>\n> (Fatou) If $(f_n)$ is any sequence ofnon-negative measurable functions de fined on the measure space $( X , { \\mathcal { M } } , \\mu )$ , then\n>\n> $$\n> \\int \\liminf _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu \\leq \\liminf _ {n \\to \\infty} \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n\n> [!proof]- Proof\n>\n> The functions $\\begin{array} { r } { G _ { n } ( x ) = \\operatorname* { i n f } _ { k \\geq n } f _ { k } ( x ) } \\end{array}$ form an increasing sequence of non-negative measurable functions such that $G _ { n } \\leq f _ { n }$ for all $n$. Hence the lim inf of the sequence $( f _ { n } )$ is the limit of the sequence $G _ { n }$\n>\n> $$\n> \\liminf _ {n \\to \\infty} f _ {n} = \\sup _ {n} G _ {n} = \\lim _ {n \\to \\infty} G _ {n}.\n> $$\n>\n> By the monotone convergence theorem 11.7,\n>\n> $$\n> \\lim _ {n \\to \\infty} \\int G _ {n} \\mathrm{d} \\mu = \\int \\operatorname * {l i m i n f} _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu\n> $$\n>\n> while the inequality $G _ { n } \\leq f _ { n }$ implies that\n>\n> $$\n> \\int G _ {n} \\mathrm{d} \\mu \\leq \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n>\n> Hence\n>\n> $$\n> \\int \\operatorname * {l i m i n f} _ {n \\to \\infty} f _ {n} \\mathrm{d} \\mu = \\lim _ {n \\to \\infty} \\int G _ {n} \\mathrm{d} \\mu \\leq \\operatorname * {l i m i n f} _ {n \\to \\infty} \\int f _ {n} \\mathrm{d} \\mu .\n> $$\n", "quote_sha256": "6391d7b0d424de28a08c5b4c81a42ee46e09d60962cfb8bbdad2c5aed61218fc", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-11/03-lebesgue-integration", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-eb53adcaceeddc44dca6291b" } ], "registered_positions": [ { "route": "en/chapter-11/03-lebesgue-integration", "lang": "en", "section": "11.3", "signal": "named-theorem", "label": "Fatou’s lemma", "english": "Fatou’s lemma", "offset": 13226 }, { "route": "zh/chapter-11/03-lebesgue-integration", "lang": "zh-CN", "section": "11.3", "signal": "named-theorem", "label": "法图引理", "english": "Fatou’s lemma", "offset": 10042 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#lemma-11-10\"", "href": "#lemma-11-10", "range": [ 16199, 16218 ], "relation": "references", "target_route": "zh/chapter-11/03-lebesgue-integration", "target_anchor": "lemma-11-10", "resolved": true, "reason": null, "source_revision": "dfa6fe6e07ccffde2e8fb24fc589de231194b6bc2966051e34615cb4e3844381", "source_route": "zh/chapter-11/03-lebesgue-integration", "occurrence_id": "occ-a7b4c1cac1e14d31671e12db" }, { "raw": "href=\"#theorem-11-7\"", "href": "#theorem-11-7", "range": [ 16815, 16835 ], "relation": "references", "target_route": "zh/chapter-11/03-lebesgue-integration", "target_anchor": "theorem-11-7", "resolved": true, "reason": null, "source_revision": "dfa6fe6e07ccffde2e8fb24fc589de231194b6bc2966051e34615cb4e3844381", "source_route": "zh/chapter-11/03-lebesgue-integration", "occurrence_id": "occ-a7b4c1cac1e14d31671e12db" }, { "raw": "href=\"#lemma-11-10\"", "href": "#lemma-11-10", "range": [ 18147, 18166 ], "relation": "references", "target_route": "en/chapter-11/03-lebesgue-integration", "target_anchor": "lemma-11-10", "resolved": true, "reason": null, "source_revision": "b10b1ac3b768247d93cf10ab40a926dfa4a8d3902771e5fca99cf3d02e26bc63", "source_route": "en/chapter-11/03-lebesgue-integration", "occurrence_id": "occ-eb53adcaceeddc44dca6291b" } ], "reading_order": [ "occ-a7b4c1cac1e14d31671e12db", "occ-eb53adcaceeddc44dca6291b" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "6328ed43bd59b744c5d6cc265331d69a46f2e6bf272308a07365dcf43827b2ec", "optional_enrichments": [] }