{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:699abcf0-77c9-4e82-a864-7d6745c8b88d", "labels": { "zh-CN": "富比尼定理", "en": "Fubini’s theorem" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "516b9f18523cfdb6e6c7921c9fb3108031e9ce33d450a0ec6ce96292202c68f8", "occurrences": [ { "concept_id": "urn:uuid:699abcf0-77c9-4e82-a864-7d6745c8b88d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-11/03-lebesgue-integration", "anchor": "first-d30daa45c714c0eae014-zh", "chapter": "11", "section": "11.3", "offset": 15247, "range": [ 20314, 21121 ], "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": "富比尼定理", "english": "Fubini’s theorem", "offset": 13676 } }, "label": "> [!theorem] 定理 11.12 · 富比尼定理 (Fubini’s theorem)", "context": { "range": [ 20314, 21121 ], "quote": "\n\n> [!theorem] 定理 11.12 · 富比尼定理 (Fubini’s theorem)\n>\n> （富比尼）若 $f:\\mathbb R^2\\to\\mathbb R$ 是勒贝格可测函数，则对每个 $x \\in \\mathbb { R }$，函数 $f _ { x } ( y ) = f ( x , y )$ 可测。类似地，对每个 $y \\in \\mathbb { R }$，函数 $f _ { y } ^ { \\prime } ( x ) = f ( x , y )$ 在 $\\mathbb R$ 上可测。通常记作\n>\n> $$\n> \\int f (x, y) \\mathrm{d} y \\equiv \\int f _ {x} \\mathrm{d} \\mu \\quad a n d \\quad \\int f (x, y) \\mathrm{d} x \\equiv \\int f _ {y} ^ {\\prime} \\mathrm{d} \\mu .\n> $$\n>\n> 于是\n>\n> $$\n> \\begin{aligned}\n> \\int \\int f (x, y) \\mathrm{d} x \\mathrm{d} y &\\equiv \\int f \\mathrm{d} \\mu^ {2} = \\int \\left(\\int f (x, y) \\mathrm{d} y\\right) \\mathrm{d} x \\\\[1em]\n> &= \\int \\left(\\int f (x, y) \\mathrm{d} x\\right) \\mathrm{d} y.\n> \\end{aligned}\n> $$\n>\n> 该结果可推广到任意一对测度空间的乘积。证明可参见例如 [1, 2]。\n", "quote_sha256": "96aee71f72798d2320ddfd61969570ef3e4eeab1862f943b5b03400449ce71af", "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-7dedf3142d92c2b6ff960ce5" }, { "concept_id": "urn:uuid:699abcf0-77c9-4e82-a864-7d6745c8b88d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-11/03-lebesgue-integration", "anchor": "first-d30daa45c714c0eae014-en", "chapter": "11", "section": "11.3", "offset": 18373, "range": [ 22968, 23965 ], "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": "Fubini’s theorem", "english": "Fubini’s theorem", "offset": 17948 } }, "label": "> [!theorem] Theorem 11.12 · Fubini’s theorem", "context": { "range": [ 22968, 23965 ], "quote": "\n\n> [!theorem] Theorem 11.12 · Fubini’s theorem\n>\n> (Fubini) If $f:\\mathbb R^2\\to\\mathbb R$ is a Lebesgue measurable function, then for each $x \\in \\mathbb { R }$ the function $f _ { x } ( y ) = f ( x , y )$ is measurable. Similarly for each $y \\in \\mathbb { R }$ the function $f _ { y } ^ { \\prime } ( x ) = f ( x , y )$ is measurable on $\\mathbb R$. It is common to write\n>\n> $$\n> \\int f (x, y) \\mathrm{d} y \\equiv \\int f _ {x} \\mathrm{d} \\mu \\quad a n d \\quad \\int f (x, y) \\mathrm{d} x \\equiv \\int f _ {y} ^ {\\prime} \\mathrm{d} \\mu .\n> $$\n>\n> Then\n>\n> $$\n> \\begin{aligned}\n> \\int \\int f (x, y) \\mathrm{d} x \\mathrm{d} y &\\equiv \\int f \\mathrm{d} \\mu^ {2} = \\int \\left(\\int f (x, y) \\mathrm{d} y\\right) \\mathrm{d} x \\\\[1em]\n> &= \\int \\left(\\int f (x, y) \\mathrm{d} x\\right) \\mathrm{d} y.\n> \\end{aligned}\n> $$\n>\n> The result generalizes to a product of an arbitrary pair of measure spaces. 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