Home | Metamath
Proof Explorer Theorem List (p. 394 of 424) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sge0ltfirpmpt 39301* | 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 39302 | 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 39303* | 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 39304* | 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 39305* | 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 39306* | 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 39307* | 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 39308* | 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 39309* | 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 39310* | 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 39311* | 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 39312* | 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 39313 | 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 39314* | 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 39315* | 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 39316* | 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 39317 | Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ℕ0 ⊆ (0[,)+∞) | ||
Theorem | sge0clmpt 39318* | 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 39319* | 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 39320 | 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 39321* | 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 39322* | 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 39323* | 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 39324* | 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 39325* | 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 39326* | 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 39327* | 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 39328* | 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 39329* | 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 39330* | 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 39331* | 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 39332* | 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 39333* | 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 39334* | 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 39335* | 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 39336* | 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 39337* | 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 39338* | 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 39339 | 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 39340* | 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 39341* | 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, < )) | ||
Proofs for most of the theorems in section 112 of [Fremlin1] | ||
Syntax | cmea 39342 | Extend class notation with the class of measures. |
class Meas | ||
Definition | df-mea 39343* | 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 39344* | 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 39345 | 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 39346 | 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 39347 | The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) ⇒ ⊢ (𝜑 → (𝑀‘∅) = 0) | ||
Theorem | nnfoctbdjlem 39348* | 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 39349* | 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 39350* | 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 39351 | 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 39352* | 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 39353* | 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 39354 | 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 39355 | 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 39356 | 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.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | meaunle 39357 | 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.) |
⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ 𝑆 = dom 𝑀 & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | ||
Theorem | meadjiunlem 39358* | 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 39359* | 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 39360* | 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 39361* | 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.) |
⊢ 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 39362* | 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.) |
⊢ 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 39363* | 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 39364* | 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 39365* | 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 39366* | 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 39367 | The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → vol ∈ Meas) | ||
Theorem | meage0 39368 | 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 39369 | 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 39370 | 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 39371 | A measure that is smaller 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 39372 | 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 39373* | 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 39374* | 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.) |
⊢ (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 39375* | 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.) |
⊢ (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 | meaiininclem 39376* | 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.) |
⊢ (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 39377* | 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.) |
⊢ 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 39378* | 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.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ Meas) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ∈ ℝ) & ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) | ||
Proofs for most of the theorems in section 113 of [Fremlin1] | ||
Syntax | come 39379 | Extend class notation with the class of outer measures. |
class OutMeas | ||
Definition | df-ome 39380* | 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 39381 | Extend class notation with a function that takes an outer measure and generates a sigma-algebra and a measure. |
class CaraGen | ||
Definition | df-caragen 39382* | 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 39383* | 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 39384* | 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 39385* | 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 39386 | 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 39387 | The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) ⇒ ⊢ (𝜑 → (𝑂‘∅) = 0) | ||
Theorem | omessle 39388 | 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.) |
⊢ (𝜑 → 𝑂 ∈ OutMeas) & ⊢ 𝑋 = ∪ dom 𝑂 & ⊢ (𝜑 → 𝐵 ⊆ 𝑋) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ≤ (𝑂‘𝐵)) | ||
Theorem | omedm 39389 | 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 39390 | 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 39391 | 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 39392* | 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 39393 | 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 39394 | 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 39395 | 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 39396 | 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 39397 | 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 39398 | 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 39399 | 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 39400 | 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) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |