{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:4b34c7bc-ab68-4690-952f-eec2de9e865c", "labels": { "zh-CN": "勒贝格控制收敛定理", "en": "Lebesgue dominated convergence theorem" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "0d536e6010002ac5c8361dab31a227a9a3a8904387d2d228092e688ccf447231", "occurrences": [ { "concept_id": "urn:uuid:4b34c7bc-ab68-4690-952f-eec2de9e865c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-11/03-lebesgue-integration", "anchor": "first-12eea7dbfea39eb77dc3-zh", "chapter": "11", "section": "11.3", "offset": 12497, "range": [ 17564, 19076 ], "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": "勒贝格控制收敛定理", "english": "Lebesgue dominated convergence theorem", "offset": 11379 } }, "label": "> [!theorem] 定理 11.11 · 勒贝格控制收敛定理 (Lebesgue dominated convergence theorem)", "context": { "range": [ 17564, 19076 ], "quote": "\n\n> [!theorem] 定理 11.11 · 勒贝格控制收敛定理 (Lebesgue dominated convergence theorem)\n>\n> （勒贝格）设 $( f _ { n } )$ 是定义在测度空间 $( X , { \\mathcal { M } } , \\mu )$ 上的任意实值可测函数序列，它几乎处处收敛于函数 $f$。若存在正的可积函数 $g : X \\to \\mathbb { R }$，使得对所有 $n$ 都有 $| f _ { n } | < g$，则\n>\n> $$\n> \\lim _ {n \\rightarrow \\infty} \\int f _ {n} d \\mu = \\int f d \\mu .\n> $$\n\n> [!proof]- 证明\n>\n> 函数 $f$ 是可测的，因为它是可测函数序列的几乎处处极限，并且由于 $| f _ { n } | < g$，所有函数 $f _ { n }$ 和 $f$ 关于测度 $\\mu$ 可积。对正可测函数序列 $g _ { n } =$ $2 g - | f _ { n } - f | > 0$ 应用法图引理 11.10\n>\n> $$\n> \\int \\liminf _ {n \\to \\infty} (2 g - | f _ {n} - f |) \\mathrm{d} \\mu \\leq \\liminf _ {n \\to \\infty} \\int (2 g - | f _ {n} - f |) \\mathrm{d} \\mu .\n> $$\n>\n> 由于 $\\textstyle \\int g \\mathrm { d } \\mu < \\infty$ 且下极限 $| f _ { n } - f | = \\operatorname* { l i m } | f _ { n } - f | = 0$，我们有\n>\n> $$\n> 0 \\leq \\operatorname * {l i m i n f} _ {n \\to \\infty} \\int - | f _ {n} - f | \\mathrm{d} \\mu = - \\operatorname * {l i m s u p} _ {n \\to \\infty} \\int | f _ {n} - f | \\mathrm{d} \\mu .\n> $$\n>\n> 由于 $\\vert f _ { n } - f \\vert > 0$，这只有在以下情况下才可能\n>\n> $$\n> \\lim _ {n \\rightarrow \\infty} \\int | f _ {n} - f | d \\mu = 0.\n> $$\n>\n> 因此\n>\n> $$\n> \\left| \\int (f _ {n} - f) \\mathrm{d} \\mu \\right| \\leq \\int | f _ {n} - f | \\mathrm{d} \\mu \\rightarrow 0,\n> $$\n>\n> 因此 $\\textstyle \\int f _ { n } \\mathrm { d } \\mu \\to \\int f \\mathrm { d } \\mu$，如所需。\n>", "quote_sha256": "8190cbdeab4da01c0c8f5c816fc2c7ace2c389d7a2b51cd484a541857c2b3c7a", "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; 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