{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-sigma-algebra-measure-space"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-sigma-algebra-measure-space-01","title":{"zh-Hant":"σ-代數與測度空間：四點載體上由原子生成的 σ-代數與一個機率測度","en":"Sigma-algebras and measure spaces: an atom-generated sigma-algebra and a probability measure on a four-point carrier"},"summary":{"zh-Hant":"在 X={0,1,2,3} 上，以原子 A_1={0}、A_2={1,2}、A_3={3} 生成的 8 個集合構成 Σ，並以 m_1=1/2、m_2=1/3、m_3=1/6 定義 μ(∪_{i∈I}A_i)=Σ_{i∈I}m_i。證明 Σ 是 σ-代數（含可數聯集的有限歸約），證明 μ 是測度（含可數可加性的有限歸約）且為機率測度；再說明 𝒢={∅,{0},{1},X} 不是 σ-代數，以及只改一個值的 ν 不是測度。","en":"On X={0,1,2,3}, the 8 sets generated by the atoms A_1={0}, A_2={1,2}, A_3={3} form Σ, and μ(∪_{i∈I}A_i)=Σ_{i∈I}m_i with m_1=1/2, m_2=1/3, m_3=1/6. Prove that Σ is a sigma-algebra (with the finite reduction of countable unions), that μ is a measure (with the finite reduction of countable additivity) and a probability measure; then show that 𝒢={∅,{0},{1},X} is not a sigma-algebra and that ν, which changes a single value, is not a measure."},"mko_ids":["mko-sigma-algebra-measure-space"]}],"title":"概念推導導讀"}