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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cdde 31601 | Extend class notation to include the Dirac delta measure. |
class δ | ||
Definition | df-dde 31602 | Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | ||
Theorem | ddeval1 31603 | Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1) | ||
Theorem | ddeval0 31604 | Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0) | ||
Theorem | ddemeas 31605 | The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ δ ∈ (measures‘𝒫 ℝ) | ||
Syntax | cae 31606 | Extend class notation to include the 'almost everywhere' relation. |
class a.e. | ||
Syntax | cfae 31607 | Extend class notation to include the 'almost everywhere' builder. |
class ~ a.e. | ||
Definition | df-ae 31608* | Define 'almost everywhere' with regard to a measure 𝑀. A property holds almost everywhere if the measure of the set where it does not hold has measure zero. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ a.e. = {〈𝑎, 𝑚〉 ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0} | ||
Theorem | relae 31609 | 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ Rel a.e. | ||
Theorem | brae 31610 | 'almost everywhere' relation for a measure and a measurable set 𝐴. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ((𝑀 ∈ ∪ ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ 𝐴)) = 0)) | ||
Theorem | braew 31611* | 'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ∪ dom 𝑀 = 𝑂 ⇒ ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)) | ||
Theorem | truae 31612* | A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ∪ dom 𝑀 = 𝑂 & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) | ||
Theorem | aean 31613* | A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ∪ dom 𝑀 = 𝑂 ⇒ ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀))) | ||
Definition | df-fae 31614* | Define a builder for an 'almost everywhere' relation between functions, from relations between function values. In this definition, the range of 𝑓 and 𝑔 is enforced in order to ensure the resulting relation is a set. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ~ a.e. = (𝑟 ∈ V, 𝑚 ∈ ∪ ran measures ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)}) | ||
Theorem | faeval 31615* | Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}) | ||
Theorem | relfae 31616 | The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀)) | ||
Theorem | brfae 31617* | 'almost everywhere' relation for two functions 𝐹 and 𝐺 with regard to the measure 𝑀. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ dom 𝑅 = 𝐷 & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀)) & ⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀)) ⇒ ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)) | ||
Syntax | cmbfm 31618 | Extend class notation with the measurable functions builder. |
class MblFnM | ||
Definition | df-mbfm 31619* |
Define the measurable function builder, which generates the set of
measurable functions from a measurable space to another one. Here, the
measurable spaces are given using their sigma-algebras 𝑠 and
𝑡,
and the spaces themselves are recovered by ∪ 𝑠 and ∪ 𝑡.
Note the similarities between the definition of measurable functions in measure theory, and of continuous functions in topology. This is the definition for the generic measure theory. For the specific case of functions from ℝ to ℂ, see df-mbf 24223. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) | ||
Theorem | ismbfm 31620* | The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24232. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) | ||
Theorem | elunirnmbfm 31621* | The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) | ||
Theorem | mbfmfun 31622 | A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | mbfmf 31623 | A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) | ||
Theorem | isanmbfm 31624 | The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) | ||
Theorem | mbfmcnvima 31625 | The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) ⇒ ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) | ||
Theorem | mbfmbfm 31626 | A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) | ||
Theorem | mbfmcst 31627* | A constant function is measurable. Cf. mbfconst 24237. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑇) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | ||
Theorem | 1stmbfm 31628 | The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (1st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆)) | ||
Theorem | 2ndmbfm 31629 | The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇)) | ||
Theorem | imambfm 31630* | 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.) |
⊢ (𝜑 → 𝐾 ∈ V) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))) | ||
Theorem | cnmbfm 31631 | A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑆 = (sigaGen‘𝐽)) & ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | ||
Theorem | mbfmco 31632 | The composition of two measurable functions is measurable. See cnmpt11 22268. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) | ||
Theorem | mbfmco2 31633* | The pair building of two measurable functions is measurable. ( cf. cnmpt1t 22270). (Contributed by Thierry Arnoux, 6-Jun-2017.) |
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇)) & ⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) ⇒ ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇))) | ||
Theorem | mbfmvolf 31634 | 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 volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ) | ||
Theorem | elmbfmvol2 31635 | 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 volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) | ||
Theorem | mbfmcnt 31636 | All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂)) | ||
Theorem | br2base 31637* | 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 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ) | ||
Theorem | dya2ub 31638 | An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ (𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅) | ||
Theorem | sxbrsigalem0 31639* | 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) | ||
Theorem | sxbrsigalem3 31640* | 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))) | ||
Theorem | dya2iocival 31641* | The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 24204. (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)))) | ||
Theorem | dya2iocress 31642* | Dyadic intervals are subsets of ℝ. (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) | ||
Theorem | dya2iocbrsiga 31643* | Dyadic intervals are Borel sets of ℝ. (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) ∈ 𝔅ℝ) | ||
Theorem | dya2icobrsiga 31644* | Dyadic intervals are Borel sets of ℝ. (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 ⊆ 𝔅ℝ | ||
Theorem | dya2icoseg 31645* | For any point and any closed-below, open-above interval of ℝ 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(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝐷)(,)(𝑋 + 𝐷)))) | ||
Theorem | dya2icoseg2 31646* | For any point and any open interval of ℝ 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 (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) | ||
Theorem | dya2iocrfn 31647* | 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) | ||
Theorem | dya2iocct 31648* | 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 | ||
Theorem | dya2iocnrect 31649* | 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) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) | ||
Theorem | dya2iocnei 31650* | 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) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) | ||
Theorem | dya2iocuni 31651* | 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 31648. (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) | ||
Theorem | dya2iocucvr 31652* | 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) | ||
Theorem | sxbrsigalem1 31653* | 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) | ||
Theorem | sxbrsigalem2 31654* | 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[,)+∞))))) | ||
Theorem | sxbrsigalem4 31655* | 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) | ||
Theorem | sxbrsigalem5 31656* | First direction for sxbrsiga 31658. (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)) ⊆ (𝔅ℝ ×s 𝔅ℝ) | ||
Theorem | sxbrsigalem6 31657 | First direction for sxbrsiga 31658, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) | ||
Theorem | sxbrsiga 31658 | The product sigma-algebra (𝔅ℝ ×s 𝔅ℝ) 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 (,)) ⇒ ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘(𝐽 ×t 𝐽)) | ||
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 31665) - it is monotone (omsmon 31666) - it is countably sub-additive (omssubadd 31668) See Definition 1.11.1 of [Bogachev] p. 41. | ||
Syntax | coms 31659 | Class declaration for the outer measure construction function. |
class toOMeas | ||
Definition | df-oms 31660* | Define a function constructing an outer measure. See omsval 31661 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 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < ))) | ||
Theorem | omsval 31661* | 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 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))) | ||
Theorem | omsfval 31662* | Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )) | ||
Theorem | omscl 31663* | A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞)) | ||
Theorem | omsf 31664 | A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 ∪ dom 𝑅⟶(0[,]+∞)) | ||
Theorem | oms0 31665 | 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) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → ∅ ∈ dom 𝑅) & ⊢ (𝜑 → (𝑅‘∅) = 0) ⇒ ⊢ (𝜑 → (𝑀‘∅) = 0) | ||
Theorem | omsmon 31666 | 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) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑄) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | omssubaddlem 31667* | 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) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑤 ∈ 𝑥(𝑅‘𝑤) < ((𝑀‘𝐴) + 𝐸)) | ||
Theorem | omssubadd 31668* | 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) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → 𝑋 ≼ ω) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴)) | ||
Syntax | ccarsg 31669 | Class declaration for the Caratheodory sigma-Algebra construction. |
class toCaraSiga | ||
Definition | df-carsg 31670* | Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 31671 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‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)}) | ||
Theorem | carsgval 31671* | Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) | ||
Theorem | carsgcl 31672 | Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) | ||
Theorem | elcarsg 31673* | Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) | ||
Theorem | baselcarsg 31674 | The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | 0elcarsg 31675 | The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsguni 31676 | The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) | ||
Theorem | elcarsgss 31677 | Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑂) | ||
Theorem | difelcarsg 31678 | The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | inelcarsg 31679* | The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | unelcarsg 31680* | The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | difelcarsg2 31681* | The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgmon 31682* | Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | carsgsigalem 31683* | Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) | ||
Theorem | fiunelcarsg 31684* | The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgclctunlem1 31685* | Lemma for carsgclctun 31689. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦))) | ||
Theorem | carsggect 31686* | The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → ¬ ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴)) | ||
Theorem | carsgclctunlem2 31687* | Lemma for carsgclctun 31689. (Contributed by Thierry Arnoux, 25-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) & ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸)) | ||
Theorem | carsgclctunlem3 31688* | Lemma for carsgclctun 31689. (Contributed by Thierry Arnoux, 24-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸)) | ||
Theorem | carsgclctun 31689* | The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgsiga 31690* | 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.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂)) | ||
Theorem | omsmeas 31691 | 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) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → ∅ ∈ dom 𝑅) & ⊢ (𝜑 → (𝑅‘∅) = 0) ⇒ ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆)) | ||
Theorem | pmeasmono 31692* | This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵)) | ||
Theorem | pmeasadd 31693* | 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.) |
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} & ⊢ (𝜑 → 𝑅 ∈ 𝑄) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) | ||
Theorem | itgeq12dv 31694* | Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥) | ||
Syntax | citgm 31695 | Extend class notation with the (measure) Bochner integral. |
class itgm | ||
Syntax | csitm 31696 | Extend class notation with the integral metric for simple functions. |
class sitm | ||
Syntax | csitg 31697 | Extend class notation with the integral of simple functions. |
class sitg | ||
Definition | df-sitg 31698* |
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 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥))))) | ||
Definition | df-sitm 31699* | Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (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*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔)))) | ||
Theorem | sitgval 31700* | 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 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ 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))))) |
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