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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dmmeasal 44401 | The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
Theorem | meaf 44402 | A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 ⇒ ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) | ||
Theorem | mea0 44403 | The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) ⇒ ⊢ (𝜑 → (𝑀‘∅) = 0) | ||
Theorem | nnfoctbdjlem 44404* | There exists a mapping from ℕ onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1)))) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛))) | ||
Theorem | nnfoctbdj 44405* | There exists a mapping from ℕ onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑋 ≼ ω) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛))) | ||
Theorem | meadjuni 44406* | The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝑋 ≼ ω) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋))) | ||
Theorem | meacl 44407 | The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) | ||
Theorem | iundjiunlem 44408* | The sets in the sequence 𝐹 are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) & ⊢ (𝜑 → 𝐽 ∈ 𝑍) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) & ⊢ (𝜑 → 𝐽 < 𝐾) ⇒ ⊢ (𝜑 → ((𝐹‘𝐽) ∩ (𝐹‘𝐾)) = ∅) | ||
Theorem | iundjiun 44409* | Given a sequence 𝐸 of sets, a sequence 𝐹 of disjoint sets is built, such that the indexed union stays the same. As in the proof of Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑛𝜑 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑉) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) ⇒ ⊢ (𝜑 → ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) ∧ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∧ Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛))) | ||
Theorem | meaxrcl 44410 | The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) | ||
Theorem | meadjun 44411 | The measure of the union of two disjoint sets is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | ||
Theorem | meassle 44412 | The measure of a set is greater than or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | meaunle 44413 | The measure of the union of two sets is less than or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | ||
Theorem | meadjiunlem 44414* | The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐺:𝑋⟶𝑆) & ⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} & ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖)) ⇒ ⊢ (𝜑 → (Σ^‘(𝑀 ↾ ran 𝐺)) = (Σ^‘(𝑀 ∘ 𝐺))) | ||
Theorem | meadjiun 44415* | The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ 𝐴 𝐵) = (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝑀‘𝐵)))) | ||
Theorem | ismeannd 44416* | Sufficient condition to prove that 𝑀 is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆 ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛))))) ⇒ ⊢ (𝜑 → 𝑀 ∈ Meas) | ||
Theorem | meaiunlelem 44417* | The measure of the union of countable sets is less than or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))))) | ||
Theorem | meaiunle 44418* | The measure of the union of countable sets is less than or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))))) | ||
Theorem | psmeasurelem 44419* | 𝑀 applied to a disjoint union of subsets of its domain is the sum of 𝑀 applied to such subset. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) & ⊢ 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥))) & ⊢ (𝜑 → 𝑀:𝒫 𝑋⟶(0[,]+∞)) & ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 𝑦) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑌) = (Σ^‘(𝑀 ↾ 𝑌))) | ||
Theorem | psmeasure 44420* | Point supported measure, Remark 112B (d) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐻:𝑋⟶(0[,]+∞)) & ⊢ 𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻 ↾ 𝑥))) ⇒ ⊢ (𝜑 → 𝑀 ∈ Meas) | ||
Theorem | voliunsge0lem 44421* | The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ 𝑆 = seq1( + , 𝐺) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) & ⊢ (𝜑 → 𝐸:ℕ⟶dom vol) & ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛)) ⇒ ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))) | ||
Theorem | voliunsge0 44422* | The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐸:ℕ⟶dom vol) & ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛)) ⇒ ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))) | ||
Theorem | volmea 44423 | The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → vol ∈ Meas) | ||
Theorem | meage0 44424 | If the measure of a measurable set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) ⇒ ⊢ (𝜑 → 0 ≤ (𝑀‘𝐴)) | ||
Theorem | meadjunre 44425 | The measure of the union of two disjoint sets, with finite measure, is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) + (𝑀‘𝐵))) | ||
Theorem | meassre 44426 | If the measure of a measurable set is real, then the measure of any of its measurable subsets is real. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) ⇒ ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) | ||
Theorem | meale0eq0 44427 | A measure that is less than or equal to 0 is 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) & ⊢ (𝜑 → (𝑀‘𝐴) ≤ 0) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) = 0) | ||
Theorem | meadif 44428 | The measure of the difference of two sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵))) | ||
Theorem | meaiuninclem 44429* | Measures are continuous from below (bounded case): if 𝐸 is a sequence of increasing measurable sets (with uniformly bounded measure) then the measure of the union is the union of the measure. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiuninc 44430* | Measures are continuous from below (bounded case): if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiuninc2 44431* | Measures are continuous from below (bounded case): if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ≤ 𝐵) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiunincf 44432* | Measures are continuous from below (bounded case): if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑛𝐸 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiuninc3v 44433* | Measures are continuous from below: if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 44430 and meaiuninc2 44431 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiuninc3 44434* | Measures are continuous from below: if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 44430 and meaiuninc2 44431 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑛𝐸 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiininclem 44435* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) & ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) & ⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiininc 44436* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) & ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Theorem | meaiininc2 44437* | Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Proofs for most of the theorems in section 113 of [Fremlin1] | ||
Syntax | come 44438 | Extend class notation with the class of outer measures. |
class OutMeas | ||
Definition | df-ome 44439* | Define the class of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))))} | ||
Syntax | ccaragen 44440 | Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure. |
class CaraGen | ||
Definition | df-caragen 44441* | Define the sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) | ||
Theorem | caragenval 44442* | The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) | ||
Theorem | isome 44443* | Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))) | ||
Theorem | caragenel 44444* | Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))) | ||
Theorem | omef 44445 | An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 ⇒ ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) | ||
Theorem | ome0 44446 | The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) ⇒ ⊢ (𝜑 → (𝑂‘∅) = 0) | ||
Theorem | omessle 44447 | The outer measure of a set is greater than or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐵 ⊆ 𝑋) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵)) | ||
Theorem | omedm 44448 | The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂) | ||
Theorem | caragensplit 44449 | If 𝐸 is in the set generated by the Caratheodory's method, then it splits any set 𝐴 in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set 𝐴. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)) | ||
Theorem | caragenelss 44450 | An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ 𝑋 = ∪ dom 𝑂 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | ||
Theorem | carageneld 44451* | Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
Theorem | omecl 44452 | The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) | ||
Theorem | caragenss 44453 | The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) | ||
Theorem | omeunile 44454 | The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) & ⊢ (𝜑 → 𝑌 ≼ ω) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))) | ||
Theorem | caragen0 44455 | The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑆) | ||
Theorem | omexrcl 44456 | The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) | ||
Theorem | caragenunidm 44457 | The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑆) | ||
Theorem | caragensspw 44458 | The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋) | ||
Theorem | omessre 44459 | If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ) | ||
Theorem | caragenuni 44460 | The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂) | ||
Theorem | caragenuncllem 44461 | The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸 ∪ 𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸 ∪ 𝐹)))) = (𝑂‘𝐴)) | ||
Theorem | caragenuncl 44462 | The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) | ||
Theorem | caragendifcl 44463 | The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | ||
Theorem | caragenfiiuncl 44464* | The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
Theorem | omeunle 44465 | The outer measure of the union of two sets is less than or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐵 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) | ||
Theorem | omeiunle 44466* | The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑛𝐸 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))) | ||
Theorem | omelesplit 44467 | The outer measure of a set 𝐴 is less than or equal to the extended addition of the outer measures of the decomposition induced on 𝐴 by any 𝐸. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) | ||
Theorem | omeiunltfirp 44468* | If the outer measure of a countable union is not +∞, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋) & ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)) | ||
Theorem | omeiunlempt 44469* | The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐸 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 𝐸) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘𝐸)))) | ||
Theorem | carageniuncllem1 44470* | The outer measure of 𝐴 ∩ (𝐺‘𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹‘𝑚). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖))) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹‘𝑛))) = (𝑂‘(𝐴 ∩ (𝐺‘𝐾)))) | ||
Theorem | carageniuncllem2 44471* | The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖))) ⇒ ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +𝑒 (𝑂‘(𝐴 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝐴) + 𝑌)) | ||
Theorem | carageniuncl 44472* | The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝑆) | ||
Theorem | caragenunicl 44473 | The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝑋 ≼ ω) ⇒ ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) | ||
Theorem | caragensal 44474 | Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
Theorem | caratheodorylem1 44475* | Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐸:𝑍⟶𝑆) & ⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛)) & ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) | ||
Theorem | caratheodorylem2 44476* | Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸:ℕ⟶𝑆) & ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛)) & ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))))) | ||
Theorem | caratheodory 44477 | Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → (𝑂 ↾ 𝑆) ∈ Meas) | ||
Theorem | 0ome 44478* | The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → 𝑂 ∈ OutMeas) | ||
Theorem | isomenndlem 44479* | 𝑂 is sub-additive w.r.t. countable indexed union, implies that 𝑂 is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑂‘∅) = 0) & ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))))) & ⊢ (𝜑 → 𝐵 ⊆ ℕ) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑌) & ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐵, (𝐹‘𝑛), ∅)) ⇒ ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))) | ||
Theorem | isomennd 44480* | Sufficient condition to prove that 𝑂 is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑂‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥) → (𝑂‘𝑦) ≤ (𝑂‘𝑥)) & ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛))))) ⇒ ⊢ (𝜑 → 𝑂 ∈ OutMeas) | ||
Theorem | caragenel2d 44481* | Membership in the Caratheodory's construction. Similar to carageneld 44451, but here "less then or equal to" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ 𝑆 = (CaraGen‘𝑂) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) ≤ (𝑂‘𝑎)) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
Theorem | omege0 44482 | If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 0 ≤ (𝑂‘𝐴)) | ||
Theorem | omess0 44483 | If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐴) = 0) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑂‘𝐵) = 0) | ||
Theorem | caragencmpl 44484 | A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐸 ⊆ 𝑋) & ⊢ (𝜑 → (𝑂‘𝐸) = 0) & ⊢ 𝑆 = (CaraGen‘𝑂) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝑆) | ||
Proofs for most of the theorems in section 115 of [Fremlin1] | ||
Syntax | covoln 44485 | Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers. |
class voln* | ||
Definition | df-ovoln 44486* | Define the outer measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑥) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )))) | ||
Syntax | cvoln 44487 | Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers. |
class voln | ||
Definition | df-voln 44488 | Define the Lebesgue measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))) | ||
Theorem | vonval 44489 | Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋)))) | ||
Theorem | ovnval 44490* | Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )))) | ||
Theorem | elhoi 44491* | Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) | ||
Theorem | icoresmbl 44492 | A closed-below, open-above real interval is measurable, when the bounds are real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol | ||
Theorem | hoissre 44493* | The projection of a half-open interval onto a single dimension is a subset of ℝ. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) | ||
Theorem | ovnval2 44494* | Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < ))) | ||
Theorem | volicorecl 44495 | The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ) | ||
Theorem | hoiprodcl 44496* | The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞)) | ||
Theorem | hoicvr 44497* | 𝐼 is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) & ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (ℝ ↑m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)) | ||
Theorem | hoissrrn 44498* | A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) ⇒ ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑m 𝑋)) | ||
Theorem | ovn0val 44499 | The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m ∅)) ⇒ ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) | ||
Theorem | ovnn0val 44500* | The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) & ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⇒ ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < )) |
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