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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsalpreimalelt 40701* If all the preimages of right-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (ii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)

Theoremsalpreimagtlt 40702* If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎 < 𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)

Theoremsmfpreimalt 40703* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremsmff 40704 A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐹:𝐷⟶ℝ)

Theoremsmfdmss 40705 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐷 𝑆)

Theoremissmff 40706* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))

Theoremissmfd 40707* 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.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmfpreimaltf 40708* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfdf 40709* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsssmf 40710 The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))       (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))

Theoremmbfresmf 40711 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑 → ran 𝐹 ⊆ ℝ)    &   𝑆 = dom vol       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremcnfsmf 40712 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.)
(𝜑𝐽 ∈ Top)    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝑆 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremincsmflem 40713* A non decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝐶 = sup(𝑌, ℝ*, < )    &   𝐷 = (-∞(,)𝐶)    &   𝐸 = (-∞(,]𝐶)       (𝜑 → ∃𝑏𝐵 𝑌 = (𝑏𝐴))

Theoremincsmf 40714* A real-valued, non-decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))

Theoremsmfsssmf 40715 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.)
(𝜑𝑅 ∈ SAlg)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑅𝑆)    &   (𝜑𝐹 ∈ (SMblFn‘𝑅))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmflelem 40716* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmfle 40717* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))

Theoremsmfpimltmpt 40718* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))

Theoremsmfpimltxr 40719* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfdmpt 40720* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵 < 𝑎} ∈ (𝑆t 𝐴))       (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))

Theoremsmfconst 40721* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsssmfmpt 40722* The restriction of a sigma-measurable function is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐶𝐴)       (𝜑 → (𝑥𝐶𝐵) ∈ (SMblFn‘𝑆))

Theoremcnfrrnsmf 40723 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.)
(𝜑𝑋 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝑋))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))

Theoremsmfid 40724* The identity function is Borel sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → (𝑥𝐴𝑥) ∈ (SMblFn‘𝐵))

Theorembormflebmf 40725 A Borel measurable function is Lebesgue measurable. Proposition 121D (a) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))    &   𝐿 = dom (voln‘𝑋)    &   (𝜑𝐹 ∈ (SMblFn‘𝐵))       (𝜑𝐹 ∈ (SMblFn‘𝐿))

Theoremsmfpreimale 40726* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfgtlem 40727* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmfgt 40728* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))))

Theoremissmfled 40729* 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.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmfpimltxrmpt 40730* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))

Theoremsmfmbfcex 40731* 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 23369). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑 → ¬ 𝑋𝑆)    &   𝐹 = (𝑥𝑋 ↦ 0)       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn))

Theoremissmfgtd 40732* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmfpreimagt 40733* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))

Theoremsmfaddlem1 40734* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 + 𝐷) < 𝑅} = 𝑝 ∈ ℚ 𝑞 ∈ (𝐾𝑝){𝑥 ∈ (𝐴𝐶) ∣ (𝐵 < 𝑝𝐷 < 𝑞)})

Theoremsmfaddlem2 40735* The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 + 𝐷) < 𝑅} ∈ (𝑆t (𝐴𝐶)))

Theoremsmfadd 40736* The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆))

Theoremdecsmflem 40737* A non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝐶 = sup(𝑌, ℝ*, < )    &   𝐷 = (-∞(,)𝐶)    &   𝐸 = (-∞(,]𝐶)       (𝜑 → ∃𝑏𝐵 𝑌 = (𝑏𝐴))

Theoremdecsmf 40738* A real-valued, non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))

Theoremsmfpreimagtf 40739* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))

Theoremissmfgelem 40740* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   (𝜑 → ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmfge 40741* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))

Theoremsmflimlem1 40742* 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 (𝐷𝐼) is in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → (𝐷𝐼) ∈ (𝑆t 𝐷))

Theoremsmflimlem2 40743* 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 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ⊆ (𝐷𝐼))

Theoremsmflimlem3 40744* The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑦 ∈ ran 𝑃) → (𝐶𝑦) ∈ 𝑦)    &   (𝜑𝑋 ∈ (𝐷𝐼))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (1 / 𝐾) < 𝑌)       (𝜑 → ∃𝑚𝑍𝑖 ∈ (ℤ𝑚)(𝑋 ∈ dom (𝐹𝑖) ∧ ((𝐹𝑖)‘𝑋) < (𝐴 + 𝑌)))

Theoremsmflimlem4 40745* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → (𝐷𝐼) ⊆ {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴})

Theoremsmflimlem5 40746* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})    &   𝐻 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))    &   𝐼 = 𝑘 ∈ ℕ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)(𝑚𝐻𝑘)    &   ((𝜑𝑟 ∈ ran 𝑃) → (𝐶𝑟) ∈ 𝑟)       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))

Theoremsmflimlem6 40747* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   𝑃 = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})       (𝜑 → {𝑥𝐷 ∣ (𝐺𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))

Theoremsmflim 40748* 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.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremnsssmfmbflem 40749* 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    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑 → ¬ 𝑋𝑆)    &   𝐹 = (𝑥𝑋 ↦ 0)       (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn))

Theoremnsssmfmbf 40750 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

Theoremsmfpimgtxr 40751* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → {𝑥𝐷𝐴 < (𝐹𝑥)} ∈ (𝑆t 𝐷))

Theoremsmfpimgtmpt 40752* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐿 < 𝐵} ∈ (𝑆t 𝐴))

Theoremsmfpreimage 40753* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷𝐴 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))

Theoremmbfpsssmf 40754 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)

Theoremsmfpimgtxrmpt 40755* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐿 < 𝐵} ∈ (𝑆t 𝐴))

Theoremsmfpimioompt 40756* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆t 𝐴))

Theoremsmfpimioo 40757 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆t 𝐷))

Theoremsmfresal 40758* Given a sigma-measurable function, the subsets of 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝑇 ∈ SAlg)

Theoremsmfrec 40759* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐶 = {𝑥𝐴𝐵 ≠ 0}       (𝜑 → (𝑥𝐶 ↦ (1 / 𝐵)) ∈ (SMblFn‘𝑆))

Theoremsmfres 40760 The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹𝐴) ∈ (SMblFn‘𝑆))

Theoremsmfmullem1 40761 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝐴)    &   𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)    &   (𝜑𝑃 ∈ ((𝑈𝑌)(,)𝑈))    &   (𝜑𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))    &   (𝜑𝑆 ∈ ((𝑉𝑌)(,)𝑉))    &   (𝜑𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))    &   (𝜑𝐻 ∈ (𝑃(,)𝑅))    &   (𝜑𝐼 ∈ (𝑆(,)𝑍))       (𝜑 → (𝐻 · 𝐼) < 𝐴)

Theoremsmfmullem2 40762* 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.)
(𝜑𝐴 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴}    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝐴)    &   (𝜑𝑃 ∈ ℚ)    &   (𝜑𝑅 ∈ ℚ)    &   (𝜑𝑆 ∈ ℚ)    &   (𝜑𝑍 ∈ ℚ)    &   (𝜑𝑃 ∈ ((𝑈𝑌)(,)𝑈))    &   (𝜑𝑅 ∈ (𝑈(,)(𝑈 + 𝑌)))    &   (𝜑𝑆 ∈ ((𝑉𝑌)(,)𝑉))    &   (𝜑𝑍 ∈ (𝑉(,)(𝑉 + 𝑌)))    &   𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)       (𝜑 → ∃𝑞𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))

Theoremsmfmullem3 40763* 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.)
(𝜑𝑅 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅}    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑉 ∈ ℝ)    &   (𝜑 → (𝑈 · 𝑉) < 𝑅)    &   𝑋 = ((𝑅 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉))))    &   𝑌 = if(1 ≤ 𝑋, 1, 𝑋)       (𝜑 → ∃𝑞𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3))))

Theoremsmfmullem4 40764* The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)    &   𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅}    &   𝐸 = (𝑞𝐾 ↦ {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))})       (𝜑 → {𝑥 ∈ (𝐴𝐶) ∣ (𝐵 · 𝐷) < 𝑅} ∈ (𝑆t (𝐴𝐶)))

Theoremsmfmul 40765* The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆))

Theoremsmfmulc1 40766* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆))

Theoremsmfdiv 40767* The fraction of two sigma-measurable functions is measurable. Proposition 121E (e) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑥𝐶) → 𝐷 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → (𝑥𝐶𝐷) ∈ (SMblFn‘𝑆))    &   𝐸 = {𝑥𝐶𝐷 ≠ 0}       (𝜑 → (𝑥 ∈ (𝐴𝐸) ↦ (𝐵 / 𝐷)) ∈ (SMblFn‘𝑆))

Theoremsmfpimbor1lem1 40768* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐺𝐽)    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝐺𝑇)

Theoremsmfpimbor1lem2 40769* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐸𝐵)    &   𝑃 = (𝐹𝐸)    &   𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (𝐹𝑒) ∈ (𝑆t 𝐷)}       (𝜑𝑃 ∈ (𝑆t 𝐷))

Theoremsmfpimbor1 40770 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐸𝐵)    &   𝑃 = (𝐹𝐸)       (𝜑𝑃 ∈ (𝑆t 𝐷))

Theoremsmf2id 40771* 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)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → (𝑥𝐴 ↦ (2 · 𝑥)) ∈ (SMblFn‘𝐵))

Theoremsmfco 40772 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐻 ∈ (SMblFn‘𝐵))       (𝜑 → (𝐻𝐹) ∈ (SMblFn‘𝑆))

Theoremsmfneg 40773* The negative of a sigma-measurable function is measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ (SMblFn‘𝑆))

Theoremsmffmpt 40774* A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))       (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)

Theoremsmflim2 40775* 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 less distinct variable restrictions than smflim and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfpimcclem 40776* Lemma for smfpimcc 40777 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑍𝑉    &   (𝜑𝑆𝑊)    &   ((𝜑𝑦 ∈ ran (𝑛𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))})) → (𝐶𝑦) ∈ 𝑦)    &   𝐻 = (𝑛𝑍 ↦ (𝐶‘{𝑠𝑆 ∣ ((𝐹𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑛))}))       (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))

Theoremsmfpimcc 40777* Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function 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.)
𝑛𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴𝐵)       (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))

Theoremissmfle2d 40778* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → (𝐹 “ (-∞(,]𝑎)) ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmflimmpt 40779* 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 indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → 𝐴𝑉)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsuplem1 40780* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐻:𝑍𝑆)    &   ((𝜑𝑛𝑍) → ((𝐹𝑛) “ (-∞(,]𝐴)) = ((𝐻𝑛) ∩ dom (𝐹𝑛)))       (𝜑 → (𝐺 “ (-∞(,]𝐴)) ∈ (𝑆t 𝐷))

Theoremsmfsuplem2 40781* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐺 “ (-∞(,]𝐴)) ∈ (𝑆t 𝐷))

Theoremsmfsuplem3 40782* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsup 40783* 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.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 ((𝐹𝑛)‘𝑥) ≤ 𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsupmpt 40784* 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.)
𝑛𝜑    &   𝑥𝜑    &   𝑦𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfsupxr 40785* 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.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ*, < ) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ sup(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ*, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinflem 40786* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinf 40787* 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.)
𝑛𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 dom (𝐹𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦 ≤ ((𝐹𝑛)‘𝑥)}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfinfmpt 40788* 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.)
𝑛𝜑    &   𝑥𝜑    &   𝑦𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑛𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵}    &   𝐺 = (𝑥𝐷 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < ))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsuplem1 40789* If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝐾𝑍)       (𝜑 → dom (𝐻𝐾) ⊆ dom (𝐹𝐾))

Theoremsmflimsuplem2 40790* 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.)
𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝑛𝑍)    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))       (𝜑𝑋 ∈ dom (𝐻𝑛))

Theoremsmflimsuplem3 40791* The limit of the (𝐻𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑 → (𝑥 ∈ {𝑥 𝑘𝑍 𝑛 ∈ (ℤ𝑘)dom (𝐻𝑛) ∣ (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Theoremsmflimsuplem4 40792* If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑𝑁𝑍)    &   (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))    &   (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )       (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)

Theoremsmflimsuplem5 40793* 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑛𝜑    &   𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑁𝑍)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))       (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))

Theoremsmflimsuplem6 40794* 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.)
𝑛𝜑    &   𝑚𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))    &   (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)    &   (𝜑𝑁𝑍)    &   (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))       (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑋)) ∈ dom ⇝ )

Theoremsmflimsuplem7 40795* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐸 = (𝑘𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑘)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑘𝑍 ↦ (𝑥 ∈ (𝐸𝑘) ↦ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑𝐷 = {𝑥 𝑛𝑍 𝑘 ∈ (ℤ𝑛)dom (𝐻𝑘) ∣ (𝑘𝑍 ↦ ((𝐻𝑘)‘𝑥)) ∈ dom ⇝ })

Theoremsmflimsuplem8 40796* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   𝐸 = (𝑘𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑘)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})    &   𝐻 = (𝑘𝑍 ↦ (𝑥 ∈ (𝐸𝑘) ↦ sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsup 40797* 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.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmflimsupmpt 40798* 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 indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfliminflem 40799* 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

Theoremsmfliminf 40800* 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.)
𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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