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Type | Label | Description |
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Statement | ||
Theorem | smfdmss 43001 | The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | ||
Theorem | issmff 43002* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
Theorem | issmfd 43003* | A sufficient condition for "𝐹 being a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfpreimaltf 43004* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | issmfdf 43005* | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | sssmf 43006 | The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆)) | ||
Theorem | mbfresmf 43007 | A real-valued measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) & ⊢ 𝑆 = dom vol ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | cnfsmf 43008 | A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) & ⊢ 𝑆 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | incsmflem 43009* | A nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} & ⊢ 𝐶 = sup(𝑌, ℝ*, < ) & ⊢ 𝐷 = (-∞(,)𝐶) & ⊢ 𝐸 = (-∞(,]𝐶) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) | ||
Theorem | incsmf 43010* | A real-valued, nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
Theorem | smfsssmf 43011 | If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑅 ∈ SAlg) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝑅 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑅)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | issmflelem 43012* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | issmfle 43013* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
Theorem | smfpimltmpt 43014* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴)) | ||
Theorem | smfpimltxr 43015* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | issmfdmpt 43016* | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ (𝑆 ↾t 𝐴)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfconst 43017* | Given a sigma-algebra over a base set X, every partial real-valued constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | sssmfmpt 43018* | The restriction of a sigma-measurable function is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | ||
Theorem | cnfrrnsmf 43019 | A function, continuous from the standard topology on the space of n-dimensional reals to the standard topology on the reals, is Borel measurable. Proposition 121D (b) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝑋)) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
Theorem | smfid 43020* | The identity function is Borel sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (SMblFn‘𝐵)) | ||
Theorem | bormflebmf 43021 | A Borel measurable function is Lebesgue measurable. Proposition 121D (a) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋))) & ⊢ 𝐿 = dom (voln‘𝑋) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐿)) | ||
Theorem | smfpreimale 43022* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded below is in the subspace sigma-algebra induced by its domain. See Proposition 121B (ii) of [Fremlin1] p. 35 (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | issmfgtlem 43023* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | issmfgt 43024* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))) | ||
Theorem | issmfled 43025* | A sufficient condition for "𝐹 being a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfpimltxrmpt 43026* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴)) | ||
Theorem | smfmbfcex 43027* | A constant function, with non-lebesgue-measurable domain is a sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) but it is not a measurable functions ( w.r.t. to df-mbf 24212). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) | ||
Theorem | issmfgtd 43028* | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfpreimagt 43029* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfaddlem1 43030* | Given the sum of two functions, the preimage of an unbounded below, open interval, expressed as the countable union of intersections of preimages of both functions. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} = ∪ 𝑝 ∈ ℚ ∪ 𝑞 ∈ (𝐾‘𝑝){𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)}) | ||
Theorem | smfaddlem2 43031* | The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) | ||
Theorem | smfadd 43032* | The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) | ||
Theorem | decsmflem 43033* | A nonincreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} & ⊢ 𝐶 = sup(𝑌, ℝ*, < ) & ⊢ 𝐷 = (-∞(,)𝐶) & ⊢ 𝐸 = (-∞(,]𝐶) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) | ||
Theorem | decsmf 43034* | A real-valued, nonincreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
Theorem | smfpreimagtf 43035* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | issmfgelem 43036* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | issmfge 43037* | The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))) | ||
Theorem | smflimlem1 43038* | Lemma for the proof that the limit of a sequence of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that (𝐷 ∩ 𝐼) is in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smflimlem2 43039* | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ (𝐷 ∩ 𝐼)) | ||
Theorem | smflimlem3 43040* | The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (𝐶‘𝑦) ∈ 𝑦) & ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∩ 𝐼)) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) & ⊢ (𝜑 → (1 / 𝐾) < 𝑌) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))) | ||
Theorem | smflimlem4 43041* | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴}) | ||
Theorem | smflimlem5 43042* | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smflimlem6 43043* | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smflim 43044* | The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | nsssmfmbflem 43045* | The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn)) | ||
Theorem | nsssmfmbf 43046 | The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol ⇒ ⊢ ¬ (SMblFn‘𝑆) ⊆ MblFn | ||
Theorem | smfpimgtxr 43047* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfpimgtmpt 43048* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) | ||
Theorem | smfpreimage 43049* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | mbfpsssmf 43050 | Real-valued measurable functions are a proper subset of sigma-measurable functions (w.r.t. the Lebesgue measure on the reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol ⇒ ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) | ||
Theorem | smfpimgtxrmpt 43051* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) | ||
Theorem | smfpimioompt 43052* | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾t 𝐴)) | ||
Theorem | smfpimioo 43053 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfresal 43054* | Given a sigma-measurable function, the subsets of ℝ whose preimage is in the sigma-algebra induced by the function's domain, form a sigma-algebra. First part of the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑇 ∈ SAlg) | ||
Theorem | smfrec 43055* | The reciprocal of a sigma-measurable functions is sigma-measurable. First part of Proposition 121E (e) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (1 / 𝐵)) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfres 43056 | The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfmullem1 43057 | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ (𝜑 → 𝐻 ∈ (𝑃(,)𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (𝑆(,)𝑍)) ⇒ ⊢ (𝜑 → (𝐻 · 𝐼) < 𝐴) | ||
Theorem | smfmullem2 43058* | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ (𝜑 → 𝑃 ∈ ℚ) & ⊢ (𝜑 → 𝑅 ∈ ℚ) & ⊢ (𝜑 → 𝑆 ∈ ℚ) & ⊢ (𝜑 → 𝑍 ∈ ℚ) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
Theorem | smfmullem3 43059* | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝑅) & ⊢ 𝑋 = ((𝑅 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
Theorem | smfmullem4 43060* | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ 𝐸 = (𝑞 ∈ 𝐾 ↦ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) | ||
Theorem | smfmul 43061* | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfmulc1 43062* | A sigma-measurable function multiplied by a constant is sigma-measurable. Proposition 121E (c) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfdiv 43063* | The fraction of two sigma-measurable functions is measurable. Proposition 121E (e) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ 𝐸 = {𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐸) ↦ (𝐵 / 𝐷)) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfpimbor1lem1 43064* | Every open set belongs to 𝑇. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐺 ∈ 𝐽) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑇) | ||
Theorem | smfpimbor1lem2 43065* | Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfpimbor1 43066 | Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smf2id 43067* | Twice the identity function is Borel sigma-measurable (just an example, to test previous general theorems). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (2 · 𝑥)) ∈ (SMblFn‘𝐵)) | ||
Theorem | smfco 43068 | The composition of a Borel sigma-measurable function with a sigma-measurable function, is sigma-measurable. Proposition 121E (g) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐻 ∈ (SMblFn‘𝐵)) ⇒ ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (SMblFn‘𝑆)) | ||
Theorem | smfneg 43069* | The negative of a sigma-measurable function is measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ (SMblFn‘𝑆)) | ||
Theorem | smffmpt 43070* | A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) | ||
Theorem | smflim2 43071* | The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). TODO: this has fewer distinct variable conditions than smflim 43044 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfpimcclem 43072* | Lemma for smfpimcc 43073 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ 𝑍 ∈ 𝑉 & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
Theorem | smfpimcc 43073* | Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function ℎ that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of 𝐴. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
Theorem | issmfle2d 43074* | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
Theorem | smflimmpt 43075* | The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). 𝐴 can contain 𝑚 as a free variable, in other words it can be thought as an indexed collection 𝐴(𝑚). 𝐵 can be thought as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfsuplem1 43076* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐻:𝑍⟶𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfsuplem2 43077* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
Theorem | smfsuplem3 43078* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfsup 43079* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfsupmpt 43080* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfsupxr 43081* | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfinflem 43082* | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfinf 43083* | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfinfmpt 43084* | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smflimsuplem1 43085* | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) ⇒ ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾)) | ||
Theorem | smflimsuplem2 43086* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑛 ∈ 𝑍) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛)) | ||
Theorem | smflimsuplem3 43087* | The limit of the (𝐻‘𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆)) | ||
Theorem | smflimsuplem4 43088* | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ≥‘𝑁)dom (𝐻‘𝑛)) & ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) | ||
Theorem | smflimsuplem5 43089* | 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) | ||
Theorem | smflimsuplem6 43090* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) | ||
Theorem | smflimsuplem7 43091* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) | ||
Theorem | smflimsuplem8 43092* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smflimsup 43093* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smflimsupmpt 43094* | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfliminflem 43095* | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfliminf 43096* | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | smfliminfmpt 43097* | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
Theorem | sigarval 43098* | Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) | ||
Theorem | sigarim 43099* | Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) ∈ ℝ) | ||
Theorem | sigarac 43100* | Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = -(𝐵𝐺𝐴)) |
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