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Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremismbfm 31801* The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24392. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
 
Theoremelunirnmbfm 31802* The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
 
Theoremmbfmfun 31803 A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝜑𝐹 ran MblFnM)       (𝜑 → Fun 𝐹)
 
Theoremmbfmf 31804 A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))       (𝜑𝐹: 𝑆 𝑇)
 
Theoremisanmbfm 31805 The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))       (𝜑𝐹 ran MblFnM)
 
Theoremmbfmcnvima 31806 The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑆MblFnM𝑇))    &   (𝜑𝐴𝑇)       (𝜑 → (𝐹𝐴) ∈ 𝑆)
 
Theoremmbfmbfm 31807 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝜑𝑀 ran measures)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))       (𝜑𝐹 ran MblFnM)
 
Theoremmbfmcst 31808* A constant function is measurable. Cf. mbfconst 24397. (Contributed by Thierry Arnoux, 26-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 = (𝑥 𝑆𝐴))    &   (𝜑𝐴 𝑇)       (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
 
Theorem1stmbfm 31809 The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
 
Theorem2ndmbfm 31810 The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)       (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
 
Theoremimambfm 31811* If the sigma-algebra in the range of a given function is generated by a collection of basic sets 𝐾, then to check the measurability of that function, we need only consider inverse images of basic sets 𝑎. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝐾 ∈ V)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 = (sigaGen‘𝐾))       (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹: 𝑆 𝑇 ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝑆)))
 
Theoremcnmbfm 31812 A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)
(𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑆 = (sigaGen‘𝐽))    &   (𝜑𝑇 = (sigaGen‘𝐾))       (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
 
Theoremmbfmco 31813 The composition of two measurable functions is measurable. See cnmpt11 22426. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝑅 ran sigAlgebra)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑅MblFnM𝑆))    &   (𝜑𝐺 ∈ (𝑆MblFnM𝑇))       (𝜑 → (𝐺𝐹) ∈ (𝑅MblFnM𝑇))
 
Theoremmbfmco2 31814* The pair building of two measurable functions is measurable. ( cf. cnmpt1t 22428). (Contributed by Thierry Arnoux, 6-Jun-2017.)
(𝜑𝑅 ran sigAlgebra)    &   (𝜑𝑆 ran sigAlgebra)    &   (𝜑𝑇 ran sigAlgebra)    &   (𝜑𝐹 ∈ (𝑅MblFnM𝑆))    &   (𝜑𝐺 ∈ (𝑅MblFnM𝑇))    &   𝐻 = (𝑥 𝑅 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
 
Theoremmbfmvolf 31815 Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
(𝐹 ∈ (dom volMblFnM𝔅) → 𝐹:ℝ⟶ℝ)
 
Theoremelmbfmvol2 31816 Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.)
(𝐹 ∈ (dom volMblFnM𝔅) → 𝐹 ∈ MblFn)
 
Theoremmbfmcnt 31817 All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
(𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑m 𝑂))
 
20.3.17.12  Borel Algebra on ` ( RR X. RR ) `
 
Theorembr2base 31818* The base set for the generator of the Borel sigma-algebra on (ℝ × ℝ) is indeed (ℝ × ℝ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
ran (𝑥 ∈ 𝔅, 𝑦 ∈ 𝔅 ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
 
Theoremdya2ub 31819 An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
(𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅)
 
Theoremsxbrsigalem0 31820* The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
 
Theoremsxbrsigalem3 31821* The sigma-algebra generated by the closed half-spaces of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed sets of (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
𝐽 = (topGen‘ran (,))       (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ⊆ (sigaGen‘(Clsd‘(𝐽 ×t 𝐽)))
 
Theoremdya2iocival 31822* The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 24364. (Contributed by Thierry Arnoux, 24-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
 
Theoremdya2iocress 31823* Dyadic intervals are subsets of . (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ⊆ ℝ)
 
Theoremdya2iocbrsiga 31824* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅)
 
Theoremdya2icobrsiga 31825* Dyadic intervals are Borel sets of . (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ran 𝐼 ⊆ 𝔅
 
Theoremdya2icoseg 31826* For any point and any closed-below, open-above interval of centered on that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑁 = (⌊‘(1 − (2 logb 𝐷)))       ((𝑋 ∈ ℝ ∧ 𝐷 ∈ ℝ+) → ∃𝑏 ∈ ran 𝐼(𝑋𝑏𝑏 ⊆ ((𝑋𝐷)(,)(𝑋 + 𝐷))))
 
Theoremdya2icoseg2 31827* For any point and any open interval of containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))       ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋𝑏𝑏𝐸))
 
Theoremdya2iocrfn 31828* The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       𝑅 Fn (ran 𝐼 × ran 𝐼)
 
Theoremdya2iocct 31829* The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       ran 𝑅 ≼ ω
 
Theoremdya2iocnrect 31830* For any point of an open rectangle in (ℝ × ℝ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    &   𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴𝐵𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
 
Theoremdya2iocnei 31831* For any point of an open set of the usual topology on (ℝ × ℝ) there is a closed-below open-above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋𝑏𝑏𝐴))
 
Theoremdya2iocuni 31832* Every open set of (ℝ × ℝ) is a union of closed-below open-above dyadic rational rectangular subsets of (ℝ × ℝ). This union must be a countable union by dya2iocct 31829. (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (𝐴 ∈ (𝐽 ×t 𝐽) → ∃𝑐 ∈ 𝒫 ran 𝑅 𝑐 = 𝐴)
 
Theoremdya2iocucvr 31833* The dyadic rectangular set collection covers (ℝ × ℝ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))        ran 𝑅 = (ℝ × ℝ)
 
Theoremsxbrsigalem1 31834* The Borel algebra on (ℝ × ℝ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4. (Contributed by Thierry Arnoux, 17-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅)
 
Theoremsxbrsigalem2 31835* The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed half-spaces of (ℝ × ℝ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
 
Theoremsxbrsigalem4 31836* The Borel algebra on (ℝ × ℝ) is generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed-below, open-above instead of open-below, closed-above in the proof as they are ultimately generated by the floor function. (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅)
 
Theoremsxbrsigalem5 31837* First direction for sxbrsiga 31839. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
𝐽 = (topGen‘ran (,))    &   𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅 ×s 𝔅)
 
Theoremsxbrsigalem6 31838 First direction for sxbrsiga 31839, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
𝐽 = (topGen‘ran (,))       (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅 ×s 𝔅)
 
Theoremsxbrsiga 31839 The product sigma-algebra (𝔅 ×s 𝔅) is the Borel algebra on (ℝ × ℝ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
𝐽 = (topGen‘ran (,))       (𝔅 ×s 𝔅) = (sigaGen‘(𝐽 ×t 𝐽))
 
20.3.17.13  Caratheodory's extension theorem

In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure 𝑅. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 31846) - it is monotone (omsmon 31847) - it is countably sub-additive (omssubadd 31849) See Definition 1.11.1 of [Bogachev] p. 41.

 
Syntaxcoms 31840 Class declaration for the outer measure construction function.
class toOMeas
 
Definitiondf-oms 31841* Define a function constructing an outer measure. See omsval 31842 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
 
Theoremomsval 31842* Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
 
Theoremomsfval 31843* Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < ))
 
Theoremomscl 31844* A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)) ⊆ (0[,]+∞))
 
Theoremomsf 31845 A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
((𝑄𝑉𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 dom 𝑅⟶(0[,]+∞))
 
Theoremoms0 31846 A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀‘∅) = 0)
 
Theoremomsmon 31847 A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 𝑄)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
Theoremomssubaddlem 31848* For any small margin 𝐸, we can find a covering approaching the outer measure of a set 𝐴 by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑𝐴 𝑄)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 𝑧𝑧 ≼ ω)}Σ*𝑤𝑥(𝑅𝑤) < ((𝑀𝐴) + 𝐸))
 
Theoremomssubadd 31849* A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   ((𝜑𝑦𝑋) → 𝐴 𝑄)    &   (𝜑𝑋 ≼ ω)       (𝜑 → (𝑀 𝑦𝑋 𝐴) ≤ Σ*𝑦𝑋(𝑀𝐴))
 
Syntaxccarsg 31850 Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
 
Definitiondf-carsg 31851* Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 31852 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)})
 
Theoremcarsgval 31852* Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
 
Theoremcarsgcl 31853 Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
 
Theoremelcarsg 31854* Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
 
Theorembaselcarsg 31855 The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑𝑂 ∈ (toCaraSiga‘𝑀))
 
Theorem0elcarsg 31856 The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
 
Theoremcarsguni 31857 The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 (toCaraSiga‘𝑀) = 𝑂)
 
Theoremelcarsgss 31858 Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑𝐴𝑂)
 
Theoremdifelcarsg 31859 The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝑂𝐴) ∈ (toCaraSiga‘𝑀))
 
Theoreminelcarsg 31860* The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremunelcarsg 31861* The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremdifelcarsg2 31862* The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgmon 31863* Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ 𝒫 𝑂)    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
Theoremcarsgsigalem 31864* Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))       ((𝜑𝑒 ∈ 𝒫 𝑂𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝑓)) ≤ ((𝑀𝑒) +𝑒 (𝑀𝑓)))
 
Theoremfiunelcarsg 31865* The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))       (𝜑 𝐴 ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgclctunlem1 31866* Lemma for carsgclctun 31870. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑Disj 𝑦𝐴 𝑦)    &   (𝜑𝐸 ∈ 𝒫 𝑂)       (𝜑 → (𝑀‘(𝐸 𝐴)) = Σ*𝑦𝐴(𝑀‘(𝐸𝑦)))
 
Theoremcarsggect 31867* The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑 → ¬ ∅ ∈ 𝐴)    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑Disj 𝑦𝐴 𝑦)    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
 
Theoremcarsgclctunlem2 31868* Lemma for carsgclctun 31870. (Contributed by Thierry Arnoux, 25-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))    &   (𝜑Disj 𝑘 ∈ ℕ 𝐴)    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))    &   (𝜑𝐸 ∈ 𝒫 𝑂)    &   (𝜑 → (𝑀𝐸) ≠ +∞)       (𝜑 → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀𝐸))
 
Theoremcarsgclctunlem3 31869* Lemma for carsgclctun 31870. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑𝐸 ∈ 𝒫 𝑂)       (𝜑 → ((𝑀‘(𝐸 𝐴)) +𝑒 (𝑀‘(𝐸 𝐴))) ≤ (𝑀𝐸))
 
Theoremcarsgclctun 31870* The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))       (𝜑 𝐴 ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgsiga 31871* The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))
 
Theoremomsmeas 31872 The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.)
𝑀 = (toOMeas‘𝑅)    &   𝑆 = (toCaraSiga‘𝑀)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀𝑆) ∈ (measures‘𝑆))
 
Theorempmeasmono 31873* This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝜑𝑃:𝑅⟶(0[,]+∞))    &   (𝜑 → (𝑃‘∅) = 0)    &   ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥𝑅Disj 𝑦𝑥 𝑦)) → (𝑃 𝑥) = Σ*𝑦𝑥(𝑃𝑦))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑 → (𝐵𝐴) ∈ 𝑅)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑃𝐴) ≤ (𝑃𝐵))
 
Theorempmeasadd 31874* A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝜑𝑃:𝑅⟶(0[,]+∞))    &   (𝜑 → (𝑃‘∅) = 0)    &   ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥𝑅Disj 𝑦𝑥 𝑦)) → (𝑃 𝑥) = Σ*𝑦𝑥(𝑃𝑦))    &   𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}    &   (𝜑𝑅𝑄)    &   (𝜑𝐴 ≼ ω)    &   ((𝜑𝑘𝐴) → 𝐵𝑅)    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → (𝑃 𝑘𝐴 𝐵) = Σ*𝑘𝐴(𝑃𝐵))
 
20.3.18  Integration
 
20.3.18.1  Lebesgue integral - misc additions
 
Theoremitgeq12dv 31875* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
20.3.18.2  Bochner integral
 
Syntaxcitgm 31876 Extend class notation with the (measure) Bochner integral.
class itgm
 
Syntaxcsitm 31877 Extend class notation with the integral metric for simple functions.
class sitm
 
Syntaxcsitg 31878 Extend class notation with the integral of simple functions.
class sitg
 
Definitiondf-sitg 31879* Define the integral of simple functions from a measurable space dom 𝑚 to a generic space 𝑤 equipped with the right scalar product. 𝑤 will later be required to be a Banach space.

These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition.

Moreover, for each 𝑥, the pre-image (𝑔 “ {𝑥}) is requested to be measurable, of finite measure.

In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra.

Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)

sitg = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)))))
 
Definitiondf-sitm 31880* Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in *, the range structure for this integral is (ℝ*𝑠s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))))
 
Theoremsitgval 31881* Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))))
 
Theoremissibf 31882* The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
 
Theoremsibf0 31883 The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ Mnd)       (𝜑 → ( dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
 
Theoremsibfmbl 31884 A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
 
Theoremsibff 31885 A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑𝐹: dom 𝑀 𝐽)
 
Theoremsibfrn 31886 A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑 → ran 𝐹 ∈ Fin)
 
Theoremsibfima 31887 Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       ((𝜑𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞))
 
Theoremsibfinima 31888 The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝐽 ∈ Fre)       (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
 
Theoremsibfof 31889 Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   𝐶 = (Base‘𝐾)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐾 ∈ TopSp)    &   (𝜑𝐽 ∈ Fre)    &   (𝜑 → ( 0 + 0 ) = (0g𝐾))       (𝜑 → (𝐹f + 𝐺) ∈ dom (𝐾sitg𝑀))
 
Theoremsitgfval 31890* Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
 
Theoremsitgclg 31891* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 31892 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   𝐺 = (Scalar‘𝑊)    &   𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ((𝜑𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥𝐵) → (𝑚 · 𝑥) ∈ 𝐵)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclbn 31892 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclcn 31893 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) = ℂfld)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclre 31894 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) = ℝfld)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitg0 31895 The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ Mnd)       (𝜑 → ((𝑊sitg𝑀)‘( dom 𝑀 × { 0 })) = 0 )
 
Theoremsitgf 31896* The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)       (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
 
Theoremsitgaddlemb 31897 Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑 → (𝑊v (𝐻 “ (0[,)+∞))) ∈ SLMod)    &   (𝜑𝐽 ∈ Fre)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &    + = (+g𝑊)       ((𝜑𝑝 ∈ ((ran 𝐹 × ran 𝐺) ∖ {⟨ 0 , 0 ⟩})) → ((𝐻‘(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))) · (2nd𝑝)) ∈ 𝐵)
 
Theoremsitmval 31898* Value of the simple function integral metric for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐷 = (dist‘𝑊)    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
 
Theoremsitmfval 31899 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐷 = (dist‘𝑊)    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))       (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹f 𝐷𝐺)))
 
Theoremsitmcl 31900 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
(𝜑𝑊 ∈ Mnd)    &   (𝜑𝑊 ∈ ∞MetSp)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))       (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) ∈ (0[,]+∞))
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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 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46009
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