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Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsmfpreimalt 43001* 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 43002 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 43003 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 43004* 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 43005* 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 43006* 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 43007* 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 43008 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 43009 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 43010 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 43011* A nondecreasing 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 43012* A real-valued, nondecreasing 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 43013 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 43014* 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 43015* 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 43016* 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 43017* 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 43018* 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 43019* 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 43020* 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 43021 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 43022* 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 43023 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 43024* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfgtlem 43025* 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 43026* 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 43027* 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 43028* 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 43029* 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 24214). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑆 = dom vol    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑 → ¬ 𝑋𝑆)    &   𝐹 = (𝑥𝑋 ↦ 0)       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn))

Theoremissmfgtd 43030* 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 43031* 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 43032* 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 43033* 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 43034* 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 43035* A nonincreasing 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 43036* A real-valued, nonincreasing 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 43037* 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 43038* 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 43039* 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 43040* 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 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.)
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 43042* 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 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 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 43044* 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 43045* 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 43046* 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 43047* 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 43048 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 43049* 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 43050* 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 43051* 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 43052 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 43053* 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 43054* 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 43055 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 43056* 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 43057* 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 43058 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 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.)
(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 43060* 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 43061* 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 43062* 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 43063* 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 43064* 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 43065* 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 43066* 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 43067* 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 43068 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 43069* 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 43070 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 43071* 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 43072* 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 43073* 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 43046 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 43074* Lemma for smfpimcc 43075 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 43075* 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 43076* 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 43077* 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.)
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 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.)
(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 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.)
(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 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.)
(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 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.)
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 43082* 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 43083* 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 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.)
(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 43085* 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 43086* 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 43087* 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 43088* 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 43089* 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 43090* 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 43091* 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 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.)
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 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.)
(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 43094* 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 43095* 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 43096* 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.)
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 43097* 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 43098* 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))

Theoremsmfliminfmpt 43099* 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.)
𝑚𝜑    &   𝑥𝜑    &   𝑛𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ}    &   𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵)))       (𝜑𝐺 ∈ (SMblFn‘𝑆))

20.38  Mathbox for Saveliy Skresanov

20.38.1  Ceva's theorem

Theoremsigarval 43100* 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)))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44899
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