P. 476 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47501-47600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sge0ssre 47501	If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ)
Theorem	sge0lefi 47502*	A sum of nonnegative extended reals is smaller than a given extended real if and only if every finite subsum is smaller than it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((Σ^‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ≤ 𝐴))
Theorem	sge0lessmpt 47503*	A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0ltfirp 47504*	If the sum of nonnegative extended reals is real, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘𝐹) < ((Σ^‘(𝐹 ↾ 𝑥)) + 𝑌))
Theorem	sge0prle 47505*	The sum of a pair of nonnegative extended reals is less than or equal their extended addition. When it is a distinct pair, than equality holds, see sge0pr 47498. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +e 𝐸))
Theorem	sge0gerpmpt 47506*	The arbitrary sum of nonnegative extended reals is greater than or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐶 ≤ ((Σ^‘(𝑥 ∈ 𝑧 ↦ 𝐵)) +e 𝑦))    ⇒   ⊢ (𝜑 → 𝐶 ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0resrnlem 47507	The sum of nonnegative extended reals restricted to the range of a function is less than or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝐴)    &   ⊢ (𝜑 → (𝐺 ↾ 𝑋):𝑋–1-1-onto→ran 𝐺)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺)))
Theorem	sge0resrn 47508	The sum of nonnegative extended reals restricted to the range of a function is less than or equal to the sum of the composition of the two functions (well-order hypothesis allows to avoid using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺)))
Theorem	sge0ssrempt 47509*	If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ∈ ℝ)
Theorem	sge0resplit 47510	Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 47513. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑈 = (𝐴 ∪ 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
Theorem	sge0le 47511*	If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
Theorem	sge0ltfirpmpt 47512*	If the extended sum of nonnegative reals is not +∞, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < ((Σ^‘(𝑥 ∈ 𝑦 ↦ 𝐵)) + 𝑌))
Theorem	sge0split 47513	Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑈 = (𝐴 ∪ 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +e (Σ^‘(𝐹 ↾ 𝐵))))
Theorem	sge0lempt 47514*	If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶)))
Theorem	sge0splitmpt 47515*	Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶)) = ((Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶)) +e (Σ^‘(𝑥 ∈ 𝐵 ↦ 𝐶))))
Theorem	sge0ss 47516*	Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))
Theorem	sge0iunmptlemfi 47517*	Sum of nonnegative extended reals over a disjoint indexed union (in this lemma, for a finite index set). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0p1 47518*	The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) = ((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +e 𝐵))
Theorem	sge0iunmptlemre 47519*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) ∈ ℝ*)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))) ∈ ℝ*)    &   ⊢ (𝜑 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶):∪ 𝑥 ∈ 𝐴 𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0fodjrnlem 47520*	Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)    &   ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)    &   ⊢ 𝑍 = (◡𝐹 “ {∅})    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
Theorem	sge0fodjrn 47521*	Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)    &   ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
Theorem	sge0iunmpt 47522*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0iun 47523*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ 𝑋 = ∪ 𝑥 ∈ 𝐴 𝐵    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵)))))
Theorem	sge0nemnf 47524	The generalized sum of nonnegative extended reals is not minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ≠ -∞)
Theorem	sge0rpcpnf 47525*	The sum of an infinite number of a positive constant, is +∞ (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
Theorem	sge0rernmpt 47526*	If the sum of nonnegative extended reals is not +∞ then no term is +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
Theorem	sge0lefimpt 47527*	A sum of nonnegative extended reals is smaller than a given extended real if and only if every finite subsum is smaller than it. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ 𝐶))
Theorem	nn0ssge0 47528	Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ℕ0 ⊆ (0[,)+∞)
Theorem	sge0clmpt 47529*	The generalized sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (0[,]+∞))
Theorem	sge0ltfirpmpt2 47530*	If the extended sum of nonnegative reals is not +∞, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ𝑥 ∈ 𝑦 𝐵 + 𝑌))
Theorem	sge0isum 47531	If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))    &   ⊢ 𝐺 = seq𝑀( + , 𝐹)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = 𝐵)
Theorem	sge0xrclmpt 47532*	The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ*)
Theorem	sge0xp 47533*	Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))) = (Σ^‘(𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐷)))
Theorem	sge0isummpt 47534*	If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = 𝐵)
Theorem	sge0ad2en 47535*	The value of the infinite geometric series 2↑ -1 + 2↑ -2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴)
Theorem	sge0isummpt2 47536*	If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = Σ𝑘 ∈ 𝑍 𝐴)
Theorem	sge0xaddlem1 47537*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) < (Σ𝑘 ∈ 𝑈 𝐵 + (𝐸 / 2)))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) < (Σ𝑘 ∈ 𝑊 𝐶 + (𝐸 / 2)))    &   ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (𝐵 + 𝐶)), ℝ*, < ) ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) + (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))) ≤ (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (𝐵 + 𝐶)), ℝ*, < ) +e 𝐸))
Theorem	sge0xaddlem2 47538*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,)+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝐵 +e 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) +e (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))))
Theorem	sge0xadd 47539*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝐵 +e 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) +e (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))))
Theorem	sge0fsummptf 47540*	The generalized sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +∞ (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	sge0snmptf 47541*	A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)
Theorem	sge0ge0mpt 47542*	The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → 0 ≤ (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0repnfmpt 47543*	The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ ¬ (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞))
Theorem	sge0pnffigtmpt 47544*	If the generalized sum of nonnegative reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)))
Theorem	sge0splitsn 47545*	Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +e 𝐷))
Theorem	sge0pnffsumgt 47546*	If the sum of nonnegative extended reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘 ∈ 𝑥 𝐵)
Theorem	sge0gtfsumgt 47547*	If the generalized sum of nonnegative reals is larger than a given number, then that number can be dominated by a finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 < (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)𝐶 < Σ𝑘 ∈ 𝑦 𝐵)
Theorem	sge0uzfsumgt 47548*	If a real number is smaller than a generalized sum of nonnegative reals, then it is smaller than some finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝐾)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 < (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 𝐶 < Σ𝑘 ∈ (𝐾...𝑚)𝐵)
Theorem	sge0pnfmpt 47549*	If a term in the sum of nonnegative extended reals is +∞, then the value of the sum is +∞. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)
Theorem	sge0seq 47550	A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))    &   ⊢ 𝐺 = seq𝑀( + , 𝐹)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Theorem	sge0reuz 47551*	Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
Theorem	sge0reuzb 47552*	Value of the generalized sum of uniformly bounded nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))
21.47.19.3  Measures Proofs for most of the theorems in section 112 of [Fremlin1]
Syntax	cmea 47553	Extend class notation with the class of measures.
class Meas
Definition	df-mea 47554*	Define the class of measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Meas = {𝑥 ∣ (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 ∈ SAlg) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥((𝑦 ≼ ω ∧ Disj 𝑤 ∈ 𝑦 𝑤) → (𝑥‘∪ 𝑦) = (Σ^‘(𝑥 ↾ 𝑦))))}
Theorem	ismea 47555*	Express the predicate "𝑀 is a measure." Definition 112A of [Fremlin1] p. 14. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
Theorem	dmmeasal 47556	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 47557	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 47558	The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    ⇒   ⊢ (𝜑 → (𝑀‘∅) = 0)
Theorem	nnfoctbdjlem 47559*	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 47560*	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 47561*	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 47562	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 47563*	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 47564*	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 47565	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 47566	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 47567	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 47568	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 47569*	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 47570*	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 47571*	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 47572*	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 47573*	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 47574*	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 47575*	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 47576*	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 47577*	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 47578	The Lebesgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → vol ∈ Meas)
Theorem	meage0 47579	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 47580	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 47581	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 47582	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 47583	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 47584*	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 47585*	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 47586*	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 47587*	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 47588*	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 47585 and meaiuninc2 47586 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 47589*	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 47585 and meaiuninc2 47586 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 47590*	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 47591*	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 47592*	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)))
21.47.19.4  Outer measures and Caratheodory's construction Proofs for most of the theorems in section 113 of [Fremlin1]
Syntax	come 47593	Extend class notation with the class of outer measures.
class OutMeas
Definition	df-ome 47594*	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 47595	Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
class CaraGen
Definition	df-caragen 47596*	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 47597*	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 47598*	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 47599*	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 47600	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[,]+∞))