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Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsge0prle 39201* 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 39194. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ (0[,]+∞))    &   (𝜑𝐸 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸))

Theoremsge0gerpmpt 39202* The arbitrary sum of nonnegative extended reals is larger 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ*)    &   ((𝜑𝑦 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐶 ≤ ((Σ^‘(𝑥𝑧𝐵)) +𝑒 𝑦))       (𝜑𝐶 ≤ (Σ^‘(𝑥𝐴𝐵)))

Theoremsge0resrnlem 39203 The sum of nonnegative extended reals restricted to the range of a function is less or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐵⟶(0[,]+∞))    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝑋 ∈ 𝒫 𝐴)    &   (𝜑 → (𝐺𝑋):𝑋1-1-onto→ran 𝐺)       (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹𝐺)))

Theoremsge0resrn 39204 The sum of nonnegative extended reals restricted to the range of a function is less 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.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐵⟶(0[,]+∞))    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝑅 We 𝐴)       (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹𝐺)))

Theoremsge0ssrempt 39205* If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)    &   (𝜑𝐶𝐴)       (𝜑 → (Σ^‘(𝑥𝐶𝐵)) ∈ ℝ)

Theoremsge0resplit 39206 Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 39209. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝑈 = (𝐴𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝐹:𝑈⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))

Theoremsge0le 39207* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝐺:𝑋⟶(0[,]+∞))    &   ((𝜑𝑥𝑋) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (Σ^𝐹) ≤ (Σ^𝐺))

Theoremsge0ltfirpmpt 39208* 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)       (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥𝐴𝐵)) < ((Σ^‘(𝑥𝑦𝐵)) + 𝑌))

Theoremsge0split 39209 Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝑈 = (𝐴𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝐹:𝑈⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))

Theoremsge0lempt 39210* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ≤ (Σ^‘(𝑥𝐴𝐶)))

Theoremsge0splitmpt 39211* Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = ∅)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥 ∈ (𝐴𝐵) ↦ 𝐶)) = ((Σ^‘(𝑥𝐴𝐶)) +𝑒^‘(𝑥𝐵𝐶))))

Theoremsge0ss 39212* Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)       (𝜑 → (Σ^‘(𝑘𝐴𝐶)) = (Σ^‘(𝑘𝐵𝐶)))

Theoremsge0iunmptlemfi 39213* 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.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → (Σ^‘(𝑘𝐵𝐶)) ∈ ℝ)       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0p1 39214* The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞))    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) = ((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒 𝐵))

Theoremsge0iunmptlemre 39215* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → (Σ^‘(𝑘𝐵𝐶)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) ∈ ℝ*)    &   (𝜑 → (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))) ∈ ℝ*)    &   (𝜑 → (𝑘 𝑥𝐴 𝐵𝐶): 𝑥𝐴 𝐵⟶(0[,]+∞))    &   (𝜑 𝑥𝐴 𝐵 ∈ V)       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0fodjrnlem 39216* 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.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶onto𝐴)    &   (𝜑Disj 𝑛𝐶 (𝐹𝑛))    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 = ∅) → 𝐵 = 0)    &   𝑍 = (𝐹 “ {∅})       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))

Theoremsge0fodjrn 39217* 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.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶onto𝐴)    &   (𝜑Disj 𝑛𝐶 (𝐹𝑛))    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 = ∅) → 𝐵 = 0)       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))

Theoremsge0iunmpt 39218* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0iun 39219* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   𝑋 = 𝑥𝐴 𝐵    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → (Σ^𝐹) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝐹𝐵)))))

Theoremsge0nemnf 39220 The generalized sum of nonnegative extended reals is not minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) ≠ -∞)

Theoremsge0rpcpnf 39221* The sum of an infinite number of a positive constant, is +∞ (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) = +∞)

Theoremsge0rernmpt 39222* If the sum of nonnegative extended reals is not +∞ then no term is +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)       ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,)+∞))

Theoremsge0lefimpt 39223* 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → ((Σ^‘(𝑥𝐴𝐵)) ≤ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥𝑦𝐵)) ≤ 𝐶))

Theoremnn0ssge0 39224 Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
0 ⊆ (0[,)+∞)

Theoremsge0clmpt 39225* The generalized sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ (0[,]+∞))

Theoremsge0ltfirpmpt2 39226* 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)       (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥𝐴𝐵)) < (Σ𝑥𝑦 𝐵 + 𝑌))

Theoremsge0isum 39227 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.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶(0[,)+∞))    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝐺𝐵)       (𝜑 → (Σ^𝐹) = 𝐵)

Theoremsge0xrclmpt 39228* The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ*)

Theoremsge0xp 39229* Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑗𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑗𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))) = (Σ^‘(𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐷)))

Theoremsge0isummpt 39230* 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.)
𝑘𝜑    &   ((𝜑𝑘𝑍) → 𝐴 ∈ (0[,)+∞))    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , (𝑘𝑍𝐴)) ⇝ 𝐵)       (𝜑 → (Σ^‘(𝑘𝑍𝐴)) = 𝐵)

Theoremsge0ad2en 39231* The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴)

Theoremsge0isummpt2 39232* 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.)
𝑘𝜑    &   ((𝜑𝑘𝑍) → 𝐴 ∈ (0[,)+∞))    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , (𝑘𝑍𝐴)) ⇝ 𝐵)       (𝜑 → (Σ^‘(𝑘𝑍𝐴)) = Σ𝑘𝑍 𝐴)

Theoremsge0xaddlem1 39233* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,)+∞))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑𝑊𝐴)    &   (𝜑𝑊 ∈ Fin)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) < (Σ𝑘𝑈 𝐵 + (𝐸 / 2)))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) < (Σ𝑘𝑊 𝐶 + (𝐸 / 2)))    &   (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑥 (𝐵 + 𝐶)), ℝ*, < ) ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) ∈ ℝ)       (𝜑 → ((Σ^‘(𝑘𝐴𝐵)) + (Σ^‘(𝑘𝐴𝐶))) ≤ (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑥 (𝐵 + 𝐶)), ℝ*, < ) +𝑒 𝐸))

Theoremsge0xaddlem2 39234* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,)+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) ∈ ℝ)       (𝜑 → (Σ^‘(𝑘𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘𝐴𝐵)) +𝑒^‘(𝑘𝐴𝐶))))

Theoremsge0xadd 39235* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘𝐴𝐵)) +𝑒^‘(𝑘𝐴𝐶))))

Theoremsge0fsummptf 39236* 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.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)

Theoremsge0snmptf 39237* A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐶 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐵 = 𝐶)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)

Theoremsge0ge0mpt 39238* The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → 0 ≤ (Σ^‘(𝑘𝐴𝐵)))

Theoremsge0repnfmpt 39239* The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → ((Σ^‘(𝑘𝐴𝐵)) ∈ ℝ ↔ ¬ (Σ^‘(𝑘𝐴𝐵)) = +∞))

Theoremsge0pnffigtmpt 39240* 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.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘𝑥𝐵)))

Theoremsge0splitsn 39241* Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘𝐴𝐶)) +𝑒 𝐷))

Theoremsge0pnffsumgt 39242* 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.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘𝑥 𝐵)

Theoremsge0gtfsumgt 39243* 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.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < (Σ^‘(𝑘𝐴𝐵)))       (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)𝐶 < Σ𝑘𝑦 𝐵)

Theoremsge0uzfsumgt 39244* 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.)
𝑘𝜑    &   (𝜑𝐾 ∈ ℤ)    &   𝑍 = (ℤ𝐾)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < (Σ^‘(𝑘𝑍𝐵)))       (𝜑 → ∃𝑚𝑍 𝐶 < Σ𝑘 ∈ (𝐾...𝑚)𝐵)

Theoremsge0pnfmpt 39245* 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.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → ∃𝑘𝐴 𝐵 = +∞)       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)

Theoremsge0seq 39246 A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶(0[,)+∞))    &   𝐺 = seq𝑀( + , 𝐹)       (𝜑 → (Σ^𝐹) = sup(ran 𝐺, ℝ*, < ))

Theoremsge0reuz 39247* Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))

Theoremsge0reuzb 39248* 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.)
𝑘𝜑    &   𝑥𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑥)       (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))

20.31.19.3  Measures

Proofs for most of the theorems in section 112 of [Fremlin1]

Syntaxcmea 39249 Extend class notation with the class of measures.
class Meas

Definitiondf-mea 39250* 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 𝑤𝑦 𝑤) → (𝑥 𝑦) = (Σ^‘(𝑥𝑦))))}

Theoremismea 39251* 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 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))

Theoremdmmeasal 39252 The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀       (𝜑𝑆 ∈ SAlg)

Theoremmeaf 39253 A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀       (𝜑𝑀:𝑆⟶(0[,]+∞))

Theoremmea0 39254 The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)       (𝜑 → (𝑀‘∅) = 0)

Theoremnnfoctbdjlem 39255* There exists a mapping from 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.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝐺:𝐴1-1-onto𝑋)    &   (𝜑Disj 𝑦𝑋 𝑦)    &   𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))       (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)))

Theoremnnfoctbdj 39256* There exists a mapping from 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.)
(𝜑𝑋 ≼ ω)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑Disj 𝑦𝑋 𝑦)       (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)))

Theoremmeadjuni 39257* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝑋𝑆)    &   (𝜑𝑋 ≼ ω)    &   (𝜑Disj 𝑥𝑋 𝑥)       (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))

Theoremmeacl 39258 The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)       (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))

Theoremiundjiunlem 39259* The sets in the sequence 𝐹 are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))    &   (𝜑𝐽𝑍)    &   (𝜑𝐾𝑍)    &   (𝜑𝐽 < 𝐾)       (𝜑 → ((𝐹𝐽) ∩ (𝐹𝐾)) = ∅)

Theoremiundjiun 39260* 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.)
𝑛𝜑    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑉)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑 → ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)(𝐹𝑛) = 𝑛 ∈ (𝑁...𝑚)(𝐸𝑛) ∧ 𝑛𝑍 (𝐹𝑛) = 𝑛𝑍 (𝐸𝑛)) ∧ Disj 𝑛𝑍 (𝐹𝑛)))

Theoremmeaxrcl 39261 The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)       (𝜑 → (𝑀𝐴) ∈ ℝ*)

Theoremmeadjun 39262 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.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) +𝑒 (𝑀𝐵)))

Theoremmeassle 39263 The measure of a set is larger or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))

Theoremmeaunle 39264 The measure of the union of two sets is less or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝑀‘(𝐴𝐵)) ≤ ((𝑀𝐴) +𝑒 (𝑀𝐵)))

Theoremmeadjiunlem 39265* The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝑋𝑉)    &   (𝜑𝐺:𝑋𝑆)    &   𝑌 = {𝑖𝑋 ∣ (𝐺𝑖) ≠ ∅}    &   (𝜑Disj 𝑖𝑋 (𝐺𝑖))       (𝜑 → (Σ^‘(𝑀 ↾ ran 𝐺)) = (Σ^‘(𝑀𝐺)))

Theoremmeadjiun 39266* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≼ ω)    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → (𝑀 𝑘𝐴 𝐵) = (Σ^‘(𝑘𝐴 ↦ (𝑀𝐵))))

Theoremismeannd 39267* Sufficient condition to prove that 𝑀 is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝑀:𝑆⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑒:ℕ⟶𝑆Disj 𝑛 ∈ ℕ (𝑒𝑛)) → (𝑀 𝑛 ∈ ℕ (𝑒𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒𝑛)))))       (𝜑𝑀 ∈ Meas)

Theoremmeaiunlelem 39268* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑆)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑 → (𝑀 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))))

Theoremmeaiunle 39269* The measure of the union of countable sets is less or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑆)       (𝜑 → (𝑀 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))))

Theorempsmeasurelem 39270* 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.)
(𝜑𝑋𝑉)    &   (𝜑𝐻:𝑋⟶(0[,]+∞))    &   𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻𝑥)))    &   (𝜑𝑀:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   (𝜑Disj 𝑦𝑌 𝑦)       (𝜑 → (𝑀 𝑌) = (Σ^‘(𝑀𝑌)))

Theorempsmeasure 39271* Point supported measure, Remark 112B (d) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐻:𝑋⟶(0[,]+∞))    &   𝑀 = (𝑥 ∈ 𝒫 𝑋 ↦ (Σ^‘(𝐻𝑥)))       (𝜑𝑀 ∈ Meas)

Theoremvoliunsge0lem 39272* The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑆 = seq1( + , 𝐺)    &   𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))    &   (𝜑𝐸:ℕ⟶dom vol)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))       (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))

Theoremvoliunsge0 39273* The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐸:ℕ⟶dom vol)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))       (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))

Theoremvolmea 39274 The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑 → vol ∈ Meas)

Theoremmeage0 39275 If the measure of a measurable set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)       (𝜑 → 0 ≤ (𝑀𝐴))

Theoremmeadjunre 39276 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.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑 → (𝑀𝐵) ∈ ℝ)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) + (𝑀𝐵)))

Theoremmeassre 39277 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ∈ dom 𝑀)       (𝜑 → (𝑀𝐵) ∈ ℝ)

Theoremmeale0eq0 39278 A measure that is smaller or equal to 0 is 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ≤ 0)       (𝜑 → (𝑀𝐴) = 0)

Theoremmeadif 39279 The measure of the difference of two sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝐴 ∈ dom 𝑀)    &   (𝜑 → (𝑀𝐴) ∈ ℝ)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) − (𝑀𝐵)))

Theoremmeaiuninclem 39280* 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 (𝑀‘(𝐸𝑛)) ≤ 𝑥)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

Theoremmeaiuninc 39281* Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 (𝑀‘(𝐸𝑛)) ≤ 𝑥)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

Theoremmeaiuninc2 39282* Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ (𝐸‘(𝑛 + 1)))    &   (𝜑𝐵 ∈ ℝ)    &   ((𝜑𝑛𝑍) → (𝑀‘(𝐸𝑛)) ≤ 𝐵)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

Theoremmeaiininclem 39283* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
(𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑𝐾 ∈ (ℤ𝑁))    &   (𝜑 → (𝑀‘(𝐸𝐾)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))    &   𝐺 = (𝑛𝑍 ↦ ((𝐸𝐾) ∖ (𝐸𝑛)))    &   𝐹 = 𝑛𝑍 (𝐺𝑛)       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

Theoremmeaiininc 39284* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
𝑛𝜑    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑𝐾 ∈ (ℤ𝑁))    &   (𝜑 → (𝑀‘(𝐸𝐾)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

Theoremmeaiininc2 39285* Measures are continuous from above: if 𝐸 is a non-increasing 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.)
𝑛𝜑    &   𝑘𝜑    &   (𝜑𝑀 ∈ Meas)    &   (𝜑𝑁 ∈ ℤ)    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶dom 𝑀)    &   ((𝜑𝑛𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸𝑛))    &   (𝜑 → ∃𝑘𝑍 (𝑀‘(𝐸𝑘)) ∈ ℝ)    &   𝑆 = (𝑛𝑍 ↦ (𝑀‘(𝐸𝑛)))       (𝜑𝑆 ⇝ (𝑀 𝑛𝑍 (𝐸𝑛)))

20.31.19.4  Outer measures and Caratheodory's construction

Proofs for most of the theorems in section 113 of [Fremlin1]

Syntaxcome 39286 Extend class notation with the class of outer measures.
class OutMeas

Definitiondf-ome 39287* 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 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))))}

Syntaxccaragen 39288 Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure.
class CaraGen

Definitiondf-caragen 39289* 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‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})

Theoremcaragenval 39290* 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‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})

Theoremisome 39291* Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑂𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))

Theoremcaragenel 39292* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))

Theoremomef 39293 An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂       (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))

Theoremome0 39294 The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)       (𝜑 → (𝑂‘∅) = 0)

Theoremomessle 39295 The outer measure of a set is larger or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐵𝑋)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))

Theoremomedm 39296 The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)

Theoremcaragensplit 39297 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐸𝑆)    &   (𝜑𝐴𝑋)       (𝜑 → ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))) = (𝑂𝐴))

Theoremcaragenelss 39298 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.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐴𝑆)    &   𝑋 = dom 𝑂       (𝜑𝐴𝑋)

Theoremcarageneld 39299* Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸 ∈ 𝒫 𝑋)    &   ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))       (𝜑𝐸𝑆)

Theoremomecl 39300 The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))

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