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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremi1frn 25201 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
 
Theoremi1fima 25202 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 β†’ (◑𝐹 β€œ 𝐴) ∈ dom vol)
 
Theoremi1fima2 25203 Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ Β¬ 0 ∈ 𝐴) β†’ (volβ€˜(◑𝐹 β€œ 𝐴)) ∈ ℝ)
 
Theoremi1fima2sn 25204 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐡 βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝐴})) ∈ ℝ)
 
Theoremi1fd 25205* A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   (πœ‘ β†’ ran 𝐹 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {π‘₯}) ∈ dom vol)    &   ((πœ‘ ∧ π‘₯ ∈ (ran 𝐹 βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {π‘₯})) ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐹 ∈ dom ∫1)
 
Theoremi1f0rn 25206 Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1 β†’ 0 ∈ ran 𝐹)
 
Theoremitg1val 25207* The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1 β†’ (∫1β€˜πΉ) = Ξ£π‘₯ ∈ (ran 𝐹 βˆ– {0})(π‘₯ Β· (volβ€˜(◑𝐹 β€œ {π‘₯}))))
 
Theoremitg1val2 25208* The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ (𝐴 ∈ Fin ∧ (ran 𝐹 βˆ– {0}) βŠ† 𝐴 ∧ 𝐴 βŠ† (ℝ βˆ– {0}))) β†’ (∫1β€˜πΉ) = Ξ£π‘₯ ∈ 𝐴 (π‘₯ Β· (volβ€˜(◑𝐹 β€œ {π‘₯}))))
 
Theoremitg1cl 25209 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 β†’ (∫1β€˜πΉ) ∈ ℝ)
 
Theoremitg1ge0 25210 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≀ 𝐹) β†’ 0 ≀ (∫1β€˜πΉ))
 
Theoremi1f0 25211 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
(ℝ Γ— {0}) ∈ dom ∫1
 
Theoremitg10 25212 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
(∫1β€˜(ℝ Γ— {0})) = 0
 
Theoremi1f1lem 25213* Lemma for i1f1 25214 and itg11 25215. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 1, 0))    β‡’   (𝐹:β„βŸΆ{0, 1} ∧ (𝐴 ∈ dom vol β†’ (◑𝐹 β€œ {1}) = 𝐴))
 
Theoremi1f1 25214* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 1, 0))    β‡’   ((𝐴 ∈ dom vol ∧ (volβ€˜π΄) ∈ ℝ) β†’ 𝐹 ∈ dom ∫1)
 
Theoremitg11 25215* The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 1, 0))    β‡’   ((𝐴 ∈ dom vol ∧ (volβ€˜π΄) ∈ ℝ) β†’ (∫1β€˜πΉ) = (volβ€˜π΄))
 
Theoremitg1addlem1 25216* Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 βŠ† (◑𝐹 β€œ {π‘˜}))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ dom vol)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (volβ€˜π΅) ∈ ℝ)    β‡’   (πœ‘ β†’ (volβ€˜βˆͺ π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (volβ€˜π΅))
 
Theoremi1faddlem 25217* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    β‡’   ((πœ‘ ∧ 𝐴 ∈ β„‚) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝐴}) = βˆͺ 𝑦 ∈ ran 𝐺((◑𝐹 β€œ {(𝐴 βˆ’ 𝑦)}) ∩ (◑𝐺 β€œ {𝑦})))
 
Theoremi1fmullem 25218* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    β‡’   ((πœ‘ ∧ 𝐴 ∈ (β„‚ βˆ– {0})) β†’ (β—‘(𝐹 ∘f Β· 𝐺) β€œ {𝐴}) = βˆͺ 𝑦 ∈ (ran 𝐺 βˆ– {0})((◑𝐹 β€œ {(𝐴 / 𝑦)}) ∩ (◑𝐺 β€œ {𝑦})))
 
Theoremi1fadd 25219 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    β‡’   (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
 
Theoremi1fmul 25220 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    β‡’   (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ dom ∫1)
 
Theoremitg1addlem2 25221* Lemma for itg1add 25226. The function 𝐼 represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both 𝑖 and 𝑗 are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 25223 and itg1addlem5 25225. (Contributed by Mario Carneiro, 26-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   πΌ = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))    β‡’   (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
 
Theoremitg1addlem3 25222* Lemma for itg1add 25226. (Contributed by Mario Carneiro, 26-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   πΌ = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))    β‡’   (((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) ∧ Β¬ (𝐴 = 0 ∧ 𝐡 = 0)) β†’ (𝐴𝐼𝐡) = (volβ€˜((◑𝐹 β€œ {𝐴}) ∩ (◑𝐺 β€œ {𝐡}))))
 
Theoremitg1addlem4 25223* Lemma for itg1add 25226. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   πΌ = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))    &   π‘ƒ = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))    β‡’   (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
 
Theoremitg1addlem4OLD 25224* Obsolete version of itg1addlem4 25223. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   πΌ = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))    &   π‘ƒ = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))    β‡’   (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
 
Theoremitg1addlem5 25225* Lemma for itg1add 25226. (Contributed by Mario Carneiro, 27-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   πΌ = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))    &   π‘ƒ = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))    β‡’   (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = ((∫1β€˜πΉ) + (∫1β€˜πΊ)))
 
Theoremitg1add 25226 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    β‡’   (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = ((∫1β€˜πΉ) + (∫1β€˜πΊ)))
 
Theoremi1fmulclem 25227 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (((πœ‘ ∧ 𝐴 β‰  0) ∧ 𝐡 ∈ ℝ) β†’ (β—‘((ℝ Γ— {𝐴}) ∘f Β· 𝐹) β€œ {𝐡}) = (◑𝐹 β€œ {(𝐡 / 𝐴)}))
 
Theoremi1fmulc 25228 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ ((ℝ Γ— {𝐴}) ∘f Β· 𝐹) ∈ dom ∫1)
 
Theoremitg1mulc 25229 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (∫1β€˜((ℝ Γ— {𝐴}) ∘f Β· 𝐹)) = (𝐴 Β· (∫1β€˜πΉ)))
 
Theoremi1fres 25230* The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside 𝐴.) (Contributed by Mario Carneiro, 29-Jun-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, (πΉβ€˜π‘₯), 0))    β‡’   ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ dom vol) β†’ 𝐺 ∈ dom ∫1)
 
Theoremi1fpos 25231* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ if(0 ≀ (πΉβ€˜π‘₯), (πΉβ€˜π‘₯), 0))    β‡’   (𝐹 ∈ dom ∫1 β†’ 𝐺 ∈ dom ∫1)
 
Theoremi1fposd 25232* Deduction form of i1fposd 25232. (Contributed by Mario Carneiro, 6-Aug-2014.)
(πœ‘ β†’ (π‘₯ ∈ ℝ ↦ 𝐴) ∈ dom ∫1)    β‡’   (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if(0 ≀ 𝐴, 𝐴, 0)) ∈ dom ∫1)
 
Theoremi1fsub 25233 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) β†’ (𝐹 ∘f βˆ’ 𝐺) ∈ dom ∫1)
 
Theoremitg1sub 25234 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) β†’ (∫1β€˜(𝐹 ∘f βˆ’ 𝐺)) = ((∫1β€˜πΉ) βˆ’ (∫1β€˜πΊ)))
 
Theoremitg10a 25235* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ (πΉβ€˜π‘₯) = 0)    β‡’   (πœ‘ β†’ (∫1β€˜πΉ) = 0)
 
Theoremitg1ge0a 25236* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ 0 ≀ (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ 0 ≀ (∫1β€˜πΉ))
 
Theoremitg1lea 25237* Approximate version of itg1le 25238. If 𝐹 ≀ 𝐺 for almost all π‘₯, then ∫1𝐹 ≀ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ (πΉβ€˜π‘₯) ≀ (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (∫1β€˜πΉ) ≀ (∫1β€˜πΊ))
 
Theoremitg1le 25238 If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≀ 𝐺) β†’ (∫1β€˜πΉ) ≀ (∫1β€˜πΊ))
 
Theoremitg1climres 25239* Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)
(πœ‘ β†’ 𝐴:β„•βŸΆdom vol)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π΄β€˜π‘›) βŠ† (π΄β€˜(𝑛 + 1)))    &   (πœ‘ β†’ βˆͺ ran 𝐴 = ℝ)    &   (πœ‘ β†’ 𝐹 ∈ dom ∫1)    &   πΊ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (π΄β€˜π‘›), (πΉβ€˜π‘₯), 0))    β‡’   (πœ‘ β†’ (𝑛 ∈ β„• ↦ (∫1β€˜πΊ)) ⇝ (∫1β€˜πΉ))
 
Theoremmbfi1fseqlem1 25240* Lemma for mbfi1fseq 25246. (Contributed by Mario Carneiro, 16-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    β‡’   (πœ‘ β†’ 𝐽:(β„• Γ— ℝ)⟢(0[,)+∞))
 
Theoremmbfi1fseqlem2 25241* Lemma for mbfi1fseq 25246. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    &   πΊ = (π‘š ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-π‘š[,]π‘š), if((π‘šπ½π‘₯) ≀ π‘š, (π‘šπ½π‘₯), π‘š), 0)))    β‡’   (𝐴 ∈ β„• β†’ (πΊβ€˜π΄) = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-𝐴[,]𝐴), if((𝐴𝐽π‘₯) ≀ 𝐴, (𝐴𝐽π‘₯), 𝐴), 0)))
 
Theoremmbfi1fseqlem3 25242* Lemma for mbfi1fseq 25246. (Contributed by Mario Carneiro, 16-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    &   πΊ = (π‘š ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-π‘š[,]π‘š), if((π‘šπ½π‘₯) ≀ π‘š, (π‘šπ½π‘₯), π‘š), 0)))    β‡’   ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ (πΊβ€˜π΄):β„βŸΆran (π‘š ∈ (0...(𝐴 Β· (2↑𝐴))) ↦ (π‘š / (2↑𝐴))))
 
Theoremmbfi1fseqlem4 25243* Lemma for mbfi1fseq 25246. This lemma is not as interesting as it is long - it is simply checking that 𝐺 is in fact a sequence of simple functions, by verifying that its range is in (0...𝑛2↑𝑛) / (2↑𝑛) (which is to say, the numbers from 0 to 𝑛 in increments of 1 / (2↑𝑛)), and also that the preimage of each point π‘˜ is measurable, because it is equal to (-𝑛[,]𝑛) ∩ (◑𝐹 β€œ (π‘˜[,)π‘˜ + 1 / (2↑𝑛))) for π‘˜ < 𝑛 and (-𝑛[,]𝑛) ∩ (◑𝐹 β€œ (π‘˜[,)+∞)) for π‘˜ = 𝑛. (Contributed by Mario Carneiro, 16-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    &   πΊ = (π‘š ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-π‘š[,]π‘š), if((π‘šπ½π‘₯) ≀ π‘š, (π‘šπ½π‘₯), π‘š), 0)))    β‡’   (πœ‘ β†’ 𝐺:β„•βŸΆdom ∫1)
 
Theoremmbfi1fseqlem5 25244* Lemma for mbfi1fseq 25246. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    &   πΊ = (π‘š ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-π‘š[,]π‘š), if((π‘šπ½π‘₯) ≀ π‘š, (π‘šπ½π‘₯), π‘š), 0)))    β‡’   ((πœ‘ ∧ 𝐴 ∈ β„•) β†’ (0𝑝 ∘r ≀ (πΊβ€˜π΄) ∧ (πΊβ€˜π΄) ∘r ≀ (πΊβ€˜(𝐴 + 1))))
 
Theoremmbfi1fseqlem6 25245* Lemma for mbfi1fseq 25246. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existential quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   π½ = (π‘š ∈ β„•, 𝑦 ∈ ℝ ↦ ((βŒŠβ€˜((πΉβ€˜π‘¦) Β· (2β†‘π‘š))) / (2β†‘π‘š)))    &   πΊ = (π‘š ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (-π‘š[,]π‘š), if((π‘šπ½π‘₯) ≀ π‘š, (π‘šπ½π‘₯), π‘š), 0)))    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔:β„•βŸΆdom ∫1 ∧ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘”β€˜π‘›) ∧ (π‘”β€˜π‘›) ∘r ≀ (π‘”β€˜(𝑛 + 1))) ∧ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘”β€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯)))
 
Theoremmbfi1fseq 25246* A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function 𝐺 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔:β„•βŸΆdom ∫1 ∧ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘”β€˜π‘›) ∧ (π‘”β€˜π‘›) ∘r ≀ (π‘”β€˜(𝑛 + 1))) ∧ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘”β€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯)))
 
Theoremmbfi1flimlem 25247* Lemma for mbfi1flim 25248. (Contributed by Mario Carneiro, 5-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆβ„)    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔:β„•βŸΆdom ∫1 ∧ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘”β€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯)))
 
Theoremmbfi1flim 25248* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ βˆƒπ‘”(𝑔:β„•βŸΆdom ∫1 ∧ βˆ€π‘₯ ∈ 𝐴 (𝑛 ∈ β„• ↦ ((π‘”β€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯)))
 
Theoremmbfmullem2 25249* Lemma for mbfmul 25251. (Contributed by Mario Carneiro, 7-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐺 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐺:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    &   (πœ‘ β†’ 𝑄:β„•βŸΆdom ∫1)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (𝑛 ∈ β„• ↦ ((π‘„β€˜π‘›)β€˜π‘₯)) ⇝ (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
 
Theoremmbfmullem 25250 Lemma for mbfmul 25251. (Contributed by Mario Carneiro, 7-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐺 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐺:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
 
Theoremmbfmul 25251 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐺 ∈ MblFn)    β‡’   (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
 
Theoremitg2lcl 25252* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {π‘₯ ∣ βˆƒπ‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 ∧ π‘₯ = (∫1β€˜π‘”))}    β‡’   πΏ βŠ† ℝ*
 
Theoremitg2val 25253* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {π‘₯ ∣ βˆƒπ‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 ∧ π‘₯ = (∫1β€˜π‘”))}    β‡’   (𝐹:β„βŸΆ(0[,]+∞) β†’ (∫2β€˜πΉ) = sup(𝐿, ℝ*, < ))
 
Theoremitg2l 25254* Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {π‘₯ ∣ βˆƒπ‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 ∧ π‘₯ = (∫1β€˜π‘”))}    β‡’   (𝐴 ∈ 𝐿 ↔ βˆƒπ‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 ∧ 𝐴 = (∫1β€˜π‘”)))
 
Theoremitg2lr 25255* Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {π‘₯ ∣ βˆƒπ‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 ∧ π‘₯ = (∫1β€˜π‘”))}    β‡’   ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≀ 𝐹) β†’ (∫1β€˜πΊ) ∈ 𝐿)
 
Theoremxrge0f 25256 A real function is a nonnegative extended real function if all its values are greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
((𝐹:β„βŸΆβ„ ∧ 0𝑝 ∘r ≀ 𝐹) β†’ 𝐹:β„βŸΆ(0[,]+∞))
 
Theoremitg2cl 25257 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:β„βŸΆ(0[,]+∞) β†’ (∫2β€˜πΉ) ∈ ℝ*)
 
Theoremitg2ub 25258 The integral of a nonnegative real function 𝐹 is an upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:β„βŸΆ(0[,]+∞) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≀ 𝐹) β†’ (∫1β€˜πΊ) ≀ (∫2β€˜πΉ))
 
Theoremitg2leub 25259* Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2β€˜πΉ), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:β„βŸΆ(0[,]+∞) ∧ 𝐴 ∈ ℝ*) β†’ ((∫2β€˜πΉ) ≀ 𝐴 ↔ βˆ€π‘” ∈ dom ∫1(𝑔 ∘r ≀ 𝐹 β†’ (∫1β€˜π‘”) ≀ 𝐴)))
 
Theoremitg2ge0 25260 The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:β„βŸΆ(0[,]+∞) β†’ 0 ≀ (∫2β€˜πΉ))
 
Theoremitg2itg1 25261 The integral of a nonnegative simple function using ∫2 is the same as its value under ∫1. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≀ 𝐹) β†’ (∫2β€˜πΉ) = (∫1β€˜πΉ))
 
Theoremitg20 25262 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∫2β€˜(ℝ Γ— {0})) = 0
 
Theoremitg2lecl 25263 If an ∫2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:β„βŸΆ(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2β€˜πΉ) ≀ 𝐴) β†’ (∫2β€˜πΉ) ∈ ℝ)
 
Theoremitg2le 25264 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:β„βŸΆ(0[,]+∞) ∧ 𝐺:β„βŸΆ(0[,]+∞) ∧ 𝐹 ∘r ≀ 𝐺) β†’ (∫2β€˜πΉ) ≀ (∫2β€˜πΊ))
 
Theoremitg2const 25265* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ dom vol ∧ (volβ€˜π΄) ∈ ℝ ∧ 𝐡 ∈ (0[,)+∞)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 𝐡, 0))) = (𝐡 Β· (volβ€˜π΄)))
 
Theoremitg2const2 25266* When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐡 ∈ ℝ+) β†’ ((volβ€˜π΄) ∈ ℝ ↔ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 𝐡, 0))) ∈ ℝ))
 
Theoremitg2seq 25267* Definitional property of the ∫2 integral: for any function 𝐹 there is a countable sequence 𝑔 of simple functions less than 𝐹 whose integrals converge to the integral of 𝐹. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 25280, but unlike that theorem this one doesn't require 𝐹 to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
(𝐹:β„βŸΆ(0[,]+∞) β†’ βˆƒπ‘”(𝑔:β„•βŸΆdom ∫1 ∧ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ∘r ≀ 𝐹 ∧ (∫2β€˜πΉ) = sup(ran (𝑛 ∈ β„• ↦ (∫1β€˜(π‘”β€˜π‘›))), ℝ*, < )))
 
Theoremitg2uba 25268* Approximate version of itg2ub 25258. If 𝐹 approximately dominates 𝐺, then ∫1𝐺 ≀ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,]+∞))    &   (πœ‘ β†’ 𝐺 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ (πΊβ€˜π‘₯) ≀ (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ (∫1β€˜πΊ) ≀ (∫2β€˜πΉ))
 
Theoremitg2lea 25269* Approximate version of itg2le 25264. If 𝐹 ≀ 𝐺 for almost all π‘₯, then ∫2𝐹 ≀ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,]+∞))    &   (πœ‘ β†’ 𝐺:β„βŸΆ(0[,]+∞))    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ (πΉβ€˜π‘₯) ≀ (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (∫2β€˜πΉ) ≀ (∫2β€˜πΊ))
 
Theoremitg2eqa 25270* Approximate equality of integrals. If 𝐹 = 𝐺 for almost all π‘₯, then ∫2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,]+∞))    &   (πœ‘ β†’ 𝐺:β„βŸΆ(0[,]+∞))    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ (vol*β€˜π΄) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ (ℝ βˆ– 𝐴)) β†’ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (∫2β€˜πΉ) = (∫2β€˜πΊ))
 
Theoremitg2mulclem 25271 Lemma for itg2mulc 25272. (Contributed by Mario Carneiro, 8-Jul-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ ℝ+)    β‡’   (πœ‘ β†’ (∫2β€˜((ℝ Γ— {𝐴}) ∘f Β· 𝐹)) ≀ (𝐴 Β· (∫2β€˜πΉ)))
 
Theoremitg2mulc 25272 The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ (0[,)+∞))    β‡’   (πœ‘ β†’ (∫2β€˜((ℝ Γ— {𝐴}) ∘f Β· 𝐹)) = (𝐴 Β· (∫2β€˜πΉ)))
 
Theoremitg2splitlem 25273* Lemma for itg2split 25274. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐴 ∈ dom vol)    &   (πœ‘ β†’ 𝐡 ∈ dom vol)    &   (πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ (0[,]+∞))    &   πΉ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 𝐢, 0))    &   πΊ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, 𝐢, 0))    &   π» = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ π‘ˆ, 𝐢, 0))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ (∫2β€˜πΊ) ∈ ℝ)    β‡’   (πœ‘ β†’ (∫2β€˜π») ≀ ((∫2β€˜πΉ) + (∫2β€˜πΊ)))
 
Theoremitg2split 25274* The ∫2 integral splits under an almost disjoint union. The proof avoids the use of itg2add 25284, which requires countable choice. (Contributed by Mario Carneiro, 11-Aug-2014.)
(πœ‘ β†’ 𝐴 ∈ dom vol)    &   (πœ‘ β†’ 𝐡 ∈ dom vol)    &   (πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ (0[,]+∞))    &   πΉ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, 𝐢, 0))    &   πΊ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, 𝐢, 0))    &   π» = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ π‘ˆ, 𝐢, 0))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ (∫2β€˜πΊ) ∈ ℝ)    β‡’   (πœ‘ β†’ (∫2β€˜π») = ((∫2β€˜πΉ) + (∫2β€˜πΊ)))
 
Theoremitg2monolem1 25275* Lemma for itg2mono 25278. We show that for any constant 𝑑 less than one, 𝑑 Β· ∫1𝐻 is less than 𝑆, and so ∫1𝐻 ≀ 𝑆, which is one half of the equality in itg2mono 25278. Consider the sequence 𝐴(𝑛) = {π‘₯ ∣ 𝑑 Β· 𝐻 ≀ 𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is ℝ, and so 𝐻 β†Ύ 𝐴(𝑛) has an integral which equals ∫1𝐻 in the limit, by itg1climres 25239. Then by taking the limit in (𝑑 Β· 𝐻) β†Ύ 𝐴(𝑛) ≀ 𝐹(𝑛), we get 𝑑 Β· ∫1𝐻 ≀ 𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)    &   π‘† = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )    &   (πœ‘ β†’ 𝑇 ∈ (0(,)1))    &   (πœ‘ β†’ 𝐻 ∈ dom ∫1)    &   (πœ‘ β†’ 𝐻 ∘r ≀ 𝐺)    &   (πœ‘ β†’ 𝑆 ∈ ℝ)    &   π΄ = (𝑛 ∈ β„• ↦ {π‘₯ ∈ ℝ ∣ (𝑇 Β· (π»β€˜π‘₯)) ≀ ((πΉβ€˜π‘›)β€˜π‘₯)})    β‡’   (πœ‘ β†’ (𝑇 Β· (∫1β€˜π»)) ≀ 𝑆)
 
Theoremitg2monolem2 25276* Lemma for itg2mono 25278. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)    &   π‘† = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )    &   (πœ‘ β†’ 𝑃 ∈ dom ∫1)    &   (πœ‘ β†’ 𝑃 ∘r ≀ 𝐺)    &   (πœ‘ β†’ Β¬ (∫1β€˜π‘ƒ) ≀ 𝑆)    β‡’   (πœ‘ β†’ 𝑆 ∈ ℝ)
 
Theoremitg2monolem3 25277* Lemma for itg2mono 25278. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)    &   π‘† = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )    &   (πœ‘ β†’ 𝑃 ∈ dom ∫1)    &   (πœ‘ β†’ 𝑃 ∘r ≀ 𝐺)    &   (πœ‘ β†’ Β¬ (∫1β€˜π‘ƒ) ≀ 𝑆)    β‡’   (πœ‘ β†’ (∫1β€˜π‘ƒ) ≀ 𝑆)
 
Theoremitg2mono 25278* The Monotone Convergence Theorem for nonnegative functions. If {(πΉβ€˜π‘›):𝑛 ∈ β„•} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2β€˜πΊ) is the limit of the sequence {(∫2β€˜(πΉβ€˜π‘›)):𝑛 ∈ β„•}. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (π‘₯ ∈ ℝ ↦ sup(ran (𝑛 ∈ β„• ↦ ((πΉβ€˜π‘›)β€˜π‘₯)), ℝ, < ))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ MblFn)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›):β„βŸΆ(0[,)+∞))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∘r ≀ (πΉβ€˜(𝑛 + 1)))    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘› ∈ β„• ((πΉβ€˜π‘›)β€˜π‘₯) ≀ 𝑦)    &   π‘† = sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(πΉβ€˜π‘›))), ℝ*, < )    β‡’   (πœ‘ β†’ (∫2β€˜πΊ) = 𝑆)
 
Theoremitg2i1fseqle 25279* Subject to the conditions coming from mbfi1fseq 25246, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    β‡’   ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (π‘ƒβ€˜π‘€) ∘r ≀ 𝐹)
 
Theoremitg2i1fseq 25280* Subject to the conditions coming from mbfi1fseq 25246, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    &   π‘† = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š)))    β‡’   (πœ‘ β†’ (∫2β€˜πΉ) = sup(ran 𝑆, ℝ*, < ))
 
Theoremitg2i1fseq2 25281* In an extension to the results of itg2i1fseq 25280, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then ∫2𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    &   π‘† = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š)))    &   (πœ‘ β†’ 𝑀 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (∫1β€˜(π‘ƒβ€˜π‘˜)) ≀ 𝑀)    β‡’   (πœ‘ β†’ 𝑆 ⇝ (∫2β€˜πΉ))
 
Theoremitg2i1fseq3 25282* Special case of itg2i1fseq2 25281: if the integral of 𝐹 is a real number, then the standard limit relation holds on the integrals of simple functions approaching 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    &   π‘† = (π‘š ∈ β„• ↦ (∫1β€˜(π‘ƒβ€˜π‘š)))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    β‡’   (πœ‘ β†’ 𝑆 ⇝ (∫2β€˜πΉ))
 
Theoremitg2addlem 25283* Lemma for itg2add 25284. (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐺 ∈ MblFn)    &   (πœ‘ β†’ 𝐺:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΊ) ∈ ℝ)    &   (πœ‘ β†’ 𝑃:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘ƒβ€˜π‘›) ∧ (π‘ƒβ€˜π‘›) ∘r ≀ (π‘ƒβ€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘ƒβ€˜π‘›)β€˜π‘₯)) ⇝ (πΉβ€˜π‘₯))    &   (πœ‘ β†’ 𝑄:β„•βŸΆdom ∫1)    &   (πœ‘ β†’ βˆ€π‘› ∈ β„• (0𝑝 ∘r ≀ (π‘„β€˜π‘›) ∧ (π‘„β€˜π‘›) ∘r ≀ (π‘„β€˜(𝑛 + 1))))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝑛 ∈ β„• ↦ ((π‘„β€˜π‘›)β€˜π‘₯)) ⇝ (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (∫2β€˜(𝐹 ∘f + 𝐺)) = ((∫2β€˜πΉ) + (∫2β€˜πΊ)))
 
Theoremitg2add 25284 The ∫2 integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐺 ∈ MblFn)    &   (πœ‘ β†’ 𝐺:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ (∫2β€˜πΊ) ∈ ℝ)    β‡’   (πœ‘ β†’ (∫2β€˜(𝐹 ∘f + 𝐺)) = ((∫2β€˜πΉ) + (∫2β€˜πΊ)))
 
Theoremitg2gt0 25285* If the function 𝐹 is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
(πœ‘ β†’ 𝐴 ∈ dom vol)    &   (πœ‘ β†’ 0 < (volβ€˜π΄))    &   (πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝐹 ∈ MblFn)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 0 < (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ 0 < (∫2β€˜πΉ))
 
Theoremitg2cnlem1 25286* Lemma for itgcn 25369. (Contributed by Mario Carneiro, 30-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    β‡’   (πœ‘ β†’ sup(ran (𝑛 ∈ β„• ↦ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((πΉβ€˜π‘₯) ≀ 𝑛, (πΉβ€˜π‘₯), 0)))), ℝ*, < ) = (∫2β€˜πΉ))
 
Theoremitg2cnlem2 25287* Lemma for itgcn 25369. (Contributed by Mario Carneiro, 31-Aug-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ Β¬ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((πΉβ€˜π‘₯) ≀ 𝑀, (πΉβ€˜π‘₯), 0))) ≀ ((∫2β€˜πΉ) βˆ’ (𝐢 / 2)))    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom vol((volβ€˜π‘’) < 𝑑 β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝑒, (πΉβ€˜π‘₯), 0))) < 𝐢))
 
Theoremitg2cn 25288* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 25561 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
(πœ‘ β†’ 𝐹:β„βŸΆ(0[,)+∞))    &   (πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ (∫2β€˜πΉ) ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ dom vol((volβ€˜π‘’) < 𝑑 β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝑒, (πΉβ€˜π‘₯), 0))) < 𝐢))
 
Theoremibllem 25289 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝐡), 𝐡, 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝐢), 𝐢, 0))
 
Theoremisibl 25290* The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of ∫𝐴𝐡 dπ‘₯, which is to say that the expression ∫𝐴𝐡 dπ‘₯ doesn't make sense unless (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐺 = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝑇), 𝑇, 0)))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 = (β„œβ€˜(𝐡 / (iβ†‘π‘˜))))    &   (πœ‘ β†’ dom 𝐹 = 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜π‘₯) = 𝐡)    β‡’   (πœ‘ β†’ (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜πΊ) ∈ ℝ)))
 
Theoremisibl2 25291* The predicate "𝐹 is integrable" when 𝐹 is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐺 = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝑇), 𝑇, 0)))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 = (β„œβ€˜(𝐡 / (iβ†‘π‘˜))))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜πΊ) ∈ ℝ)))
 
Theoremiblmbf 25292 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐹 ∈ 𝐿1 β†’ 𝐹 ∈ MblFn)
 
Theoremiblitg 25293* If a function is integrable, then the ∫2 integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(πœ‘ β†’ 𝐺 = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝑇), 𝑇, 0)))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝑇 = (β„œβ€˜(𝐡 / (i↑𝐾))))    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ 𝐿1)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    β‡’   ((πœ‘ ∧ 𝐾 ∈ β„€) β†’ (∫2β€˜πΊ) ∈ ℝ)
 
Theoremdfitg 25294* Evaluate the class substitution in df-itg 25147. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝑇 = (β„œβ€˜(𝐡 / (iβ†‘π‘˜)))    β‡’   βˆ«π΄π΅ dπ‘₯ = Ξ£π‘˜ ∈ (0...3)((iβ†‘π‘˜) Β· (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ 𝑇), 𝑇, 0))))
 
Theoremitgex 25295 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
∫𝐴𝐡 dπ‘₯ ∈ V
 
Theoremitgeq1f 25296 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   (𝐴 = 𝐡 β†’ ∫𝐴𝐢 dπ‘₯ = ∫𝐡𝐢 dπ‘₯)
 
Theoremitgeq1 25297* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 = 𝐡 β†’ ∫𝐴𝐢 dπ‘₯ = ∫𝐡𝐢 dπ‘₯)
 
Theoremnfitg1 25298 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
β„²π‘₯∫𝐴𝐡 dπ‘₯
 
Theoremnfitg 25299* Bound-variable hypothesis builder for an integral: if 𝑦 is (effectively) not free in 𝐴 and 𝐡, it is not free in ∫𝐴𝐡 dπ‘₯. (Contributed by Mario Carneiro, 28-Jun-2014.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦βˆ«π΄π΅ dπ‘₯
 
Theoremcbvitg 25300* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    &   β„²π‘¦π΅    &   β„²π‘₯𝐢    β‡’   βˆ«π΄π΅ dπ‘₯ = ∫𝐴𝐢 d𝑦
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