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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sxuni 30801 | The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.) |
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇)) | ||
Theorem | elsx 30802 | The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇)) | ||
Syntax | cmeas 30803 | Extend class notation to include the class of measures. |
class measures | ||
Definition | df-meas 30804* | Define a measure as a nonnegative countably additive function over a sigma-algebra onto (0[,]+∞). (Contributed by Thierry Arnoux, 10-Sep-2016.) |
⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))}) | ||
Theorem | measbase 30805 | The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | ||
Theorem | measval 30806* | The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.) |
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (measures‘𝑆) = {𝑚 ∣ (𝑚:𝑆⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))}) | ||
Theorem | ismeas 30807* | The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.) |
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))) | ||
Theorem | isrnmeas 30808* | The property of being a measure on an undefined base sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑀 ∈ ∪ ran measures → (dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))) | ||
Theorem | dmmeas 30809 | The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra) | ||
Theorem | measbasedom 30810 | The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | ||
Theorem | measfrge0 30811 | A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞)) | ||
Theorem | measfn 30812 | A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.) |
⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆) | ||
Theorem | measvxrge0 30813 | The values of a measure are positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | ||
Theorem | measvnul 30814 | The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) | ||
Theorem | measge0 30815 | A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → 0 ≤ (𝑀‘𝐴)) | ||
Theorem | measle0 30816 | If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ≤ 0) → (𝑀‘𝐴) = 0) | ||
Theorem | measvun 30817* | The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑀‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑀‘𝑥)) | ||
Theorem | measxun2 30818 | The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐴) = ((𝑀‘𝐵) +𝑒 (𝑀‘(𝐴 ∖ 𝐵)))) | ||
Theorem | measun 30819 | The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | ||
Theorem | measvunilem 30820* | Lemma for measvuni 30822. (Contributed by Thierry Arnoux, 7-Feb-2017.) (Revised by Thierry Arnoux, 19-Feb-2017.) (Revised by Thierry Arnoux, 6-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑆 ∖ {∅}) ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵)) | ||
Theorem | measvunilem0 30821* | Lemma for measvuni 30822. (Contributed by Thierry Arnoux, 6-Mar-2017.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ {∅} ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵)) | ||
Theorem | measvuni 30822* | The measure of a countable disjoint union is the sum of the measures. This theorem uses a collection rather than a set of subsets of 𝑆. (Contributed by Thierry Arnoux, 7-Mar-2017.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ*𝑥 ∈ 𝐴(𝑀‘𝐵)) | ||
Theorem | measssd 30823 | A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | measunl 30824 | A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | ||
Theorem | measiuns 30825* | The measure of the union of a collection of sets, expressed as the sum of a disjoint set. This is used as a lemma for both measiun 30826 and meascnbl 30827. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.) |
⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼))) & ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) | ||
Theorem | measiun 30826* | A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.) |
⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) | ||
Theorem | meascnbl 30827* | A measure is continuous from below. Cf. volsup 23722. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.) |
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) & ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) ⇒ ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝𝑡‘𝐽)(𝑀‘∪ ran 𝐹)) | ||
Theorem | measinblem 30828* | Lemma for measinb 30829. (Contributed by Thierry Arnoux, 2-Jun-2017.) |
⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) ∧ 𝐵 ∈ 𝒫 𝑆) ∧ (𝐵 ≼ ω ∧ Disj 𝑥 ∈ 𝐵 𝑥)) → (𝑀‘(∪ 𝐵 ∩ 𝐴)) = Σ*𝑥 ∈ 𝐵(𝑀‘(𝑥 ∩ 𝐴))) | ||
Theorem | measinb 30829* | Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘𝑆)) | ||
Theorem | measres 30830 | Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇)) | ||
Theorem | measinb2 30831* | Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑥 ∈ (𝑆 ∩ 𝒫 𝐴) ↦ (𝑀‘(𝑥 ∩ 𝐴))) ∈ (measures‘(𝑆 ∩ 𝒫 𝐴))) | ||
Theorem | measdivcstOLD 30832* | Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 𝐴)) ∈ (measures‘𝑆)) | ||
Theorem | measdivcst 30833 | Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (Revised by Thierry Arnoux, 30-Jan-2017.) |
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ+) → (𝑀∘𝑓/𝑐 /𝑒 𝐴) ∈ (measures‘𝑆)) | ||
Theorem | cntmeas 30834 | The Counting measure is a measure on any sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (♯ ↾ 𝑆) ∈ (measures‘𝑆)) | ||
Theorem | pwcntmeas 30835 | The counting measure is a measure on any power set. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝑂 ∈ 𝑉 → (♯ ↾ 𝒫 𝑂) ∈ (measures‘𝒫 𝑂)) | ||
Theorem | cntnevol 30836 | Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.) |
⊢ (♯ ↾ 𝒫 𝑂) ≠ vol | ||
Theorem | voliune 30837 | The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for +∞ for the measure of any set in the sum. Cf. ovoliun 23671 and voliun 23720. (Contributed by Thierry Arnoux, 16-Oct-2017.) |
⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴)) | ||
Theorem | volfiniune 30838* | The Lebesgue measure function is countably additive. This theorem is to volfiniun 23713 what voliune 30837 is to voliun 23720. (Contributed by Thierry Arnoux, 16-Oct-2017.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ*𝑛 ∈ 𝐴(vol‘𝐵)) | ||
Theorem | volmeas 30839 | The Lebesgue measure is a measure. (Contributed by Thierry Arnoux, 16-Oct-2017.) |
⊢ vol ∈ (measures‘dom vol) | ||
Syntax | cdde 30840 | Extend class notation to include the Dirac delta measure. |
class δ | ||
Definition | df-dde 30841 | Define the Dirac delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | ||
Theorem | ddeval1 30842 | Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1) | ||
Theorem | ddeval0 30843 | Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0) | ||
Theorem | ddemeas 30844 | The Dirac delta measure is a measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
⊢ δ ∈ (measures‘𝒫 ℝ) | ||
Syntax | cae 30845 | Extend class notation to include the 'almost everywhere' relation. |
class a.e. | ||
Syntax | cfae 30846 | Extend class notation to include the 'almost everywhere' builder. |
class ~ a.e. | ||
Definition | df-ae 30847* | 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 30848 | 'almost everywhere' is a relation. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ Rel a.e. | ||
Theorem | brae 30849 | '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 30850* | '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 30851* | 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 30852* | 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 30853* | 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 30854* | 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 30855 | 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 30856* | '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 30857 | Extend class notation with the measurable functions builder. |
class MblFnM | ||
Definition | df-mbfm 30858* |
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 23785. (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 30859* | The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23794. (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 30860* | 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 30861 | A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | mbfmf 30862 | 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 30863 | 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 30864 | 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 30865 | 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 30866* | A constant function is measurable. Cf. mbfconst 23799. (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 30867 | 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 30868 | 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 30869* | 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 30870 | 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 30871 | The composition of two measurable functions is measurable. ( cf. cnmpt11 21837) (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 30872* | The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21839). (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 30873 | 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 30874 | 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 30875 | 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 30876* | 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 30877 | 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 30878* | 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 30879* | 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 30880* | The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23766. (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 30881* | 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 30882* | 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 30883* | 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 30884* | 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 30885* | 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 30886* | 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 30887* | 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 30888* | 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 30889* | 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 30890* | 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 30887. (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 30891* | 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 30892* | 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 30893* | 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 30894* | 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 30895* | First direction for sxbrsiga 30897. (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 30896 | First direction for sxbrsiga 30897, 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 30897 | 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 30904) - it is monotone (omsmon 30905) - it is countably sub-additive (omssubadd 30907) See Definition 1.11.1 of [Bogachev] p. 41. | ||
Syntax | coms 30898 | Class declaration for the outer measure construction function. |
class toOMeas | ||
Definition | df-oms 30899* | Define a function constructing an outer measure. See omsval 30900 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 30900* | 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[,]+∞), < ))) |
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