P. 417 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41601-41700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iundjiun 41601*	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 41602	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 41603	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 41604	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 41605	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 41606*	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 41607*	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 41608*	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 41609*	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 41610*	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 41611*	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 41612*	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 41613*	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 41614*	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 41615	The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → vol ∈ Meas)
Theorem	meage0 41616	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 41617	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 41618	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 41619	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 41620	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 41621*	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 41622*	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 41623*	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 41624*	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 41625*	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 41622 and meaiuninc2 41623 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 41626*	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 41622 and meaiuninc2 41623 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 41627*	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 41628*	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 41629*	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)))
20.33.19.4  Outer measures and Caratheodory's construction Proofs for most of the theorems in section 113 of [Fremlin1]
Syntax	come 41630	Extend class notation with the class of outer measures.
class OutMeas
Definition	df-ome 41631*	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 41632	Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
class CaraGen
Definition	df-caragen 41633*	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 41634*	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 41635*	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 41636*	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 41637	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 41638	The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑂 ∈ OutMeas)    ⇒   ⊢ (𝜑 → (𝑂‘∅) = 0)
Theorem	omessle 41639	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 41640	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 41641	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 41642	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 41643*	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 41644	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 41645	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 41646	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 41647	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 41648	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 41649	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 41650	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 41651	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 41652	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 41653	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 41654	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 41655	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 41656*	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 41657	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 41658*	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 41659	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 41660*	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 41661*	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 41662*	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 41663*	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 41664*	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 41665	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 41666	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 41667*	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 41668*	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 41669	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 41670*	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 41671*	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 41672*	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 41673*	Membership in the Caratheodory's construction. Similar to carageneld 41643, 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 41674	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 41675	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 41676	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)
20.33.19.5  Lebesgue measure on n-dimensional Real numbers Proofs for most of the theorems in section 115 of [Fremlin1]
Syntax	covoln 41677	Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
class voln*
Definition	df-ovoln 41678*	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 41679	Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
class voln
Definition	df-voln 41680	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 41681	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 41682*	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 41683*	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 41684	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 41685*	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 41686*	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 41687	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 41688*	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 41689*	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 41690*	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 41691	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 41692*	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*, < ))
Theorem	ovnval2b 41693*	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))    &   ⊢ 𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {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((𝐿‘𝐴), ℝ*, < )))
Theorem	volicorescl 41694	The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) → (vol‘𝐴) ∈ ℝ)
Theorem	ovnprodcl 41695*	The product used in the definition of the outer Lebesgue measure in R^n is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐹‘𝐼))‘𝑘)) ∈ (0[,)+∞))
Theorem	hoiprodcl2 41696*	The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))    ⇒   ⊢ (𝜑 → (𝐿‘𝐼) ∈ (0[,)+∞))
Theorem	hoicvrrex 41697*	Any subset of the multidimensional reals can be covered by a countable set of half-open intervals, see Definition 115A (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑𝑚 𝑋))    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
Theorem	ovnsupge0 41698*	The set used in the definition of the Lebesgue outer measure is a subset of the nonnegative extended reals. This is a substep for (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (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 → 𝑀 ⊆ (0[,]+∞))
Theorem	ovnlecvr 41699*	Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with 𝑋 the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
Theorem	ovnpnfelsup 41700*	+∞ is an element of the set used in the definition of the Lebesgue outer measure. (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 → +∞ ∈ 𝑀)