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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0pval 24201 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
 
Theorem0plef 24202 Two ways to say that the function 𝐹 on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝r𝐹))
 
Theorem0pledm 24203 Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹 Fn 𝐴)       (𝜑 → (0𝑝r𝐹 ↔ (𝐴 × {0}) ∘r𝐹))
 
Theoremisi1f 24204 The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because 1 is the first preparation function for our final definition (see df-itg 24153); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
 
Theoremi1fmbf 24205 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
 
Theoremi1ff 24206 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
 
Theoremi1frn 24207 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
 
Theoremi1fima 24208 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (𝐹𝐴) ∈ dom vol)
 
Theoremi1fima2 24209 Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴) → (vol‘(𝐹𝐴)) ∈ ℝ)
 
Theoremi1fima2sn 24210 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(𝐹 “ {𝐴})) ∈ ℝ)
 
Theoremi1fd 24211* A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑 → ran 𝐹 ∈ Fin)    &   ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)    &   ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)       (𝜑𝐹 ∈ dom ∫1)
 
Theoremi1f0rn 24212 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 → 0 ∈ ran 𝐹)
 
Theoremitg1val 24213* The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(𝐹 “ {𝑥}))))
 
Theoremitg1val2 24214* The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ (𝐴 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ 𝐴𝐴 ⊆ (ℝ ∖ {0}))) → (∫1𝐹) = Σ𝑥𝐴 (𝑥 · (vol‘(𝐹 “ {𝑥}))))
 
Theoremitg1cl 24215 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (∫1𝐹) ∈ ℝ)
 
Theoremitg1ge0 24216 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝r𝐹) → 0 ≤ (∫1𝐹))
 
Theoremi1f0 24217 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
(ℝ × {0}) ∈ dom ∫1
 
Theoremitg10 24218 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
(∫1‘(ℝ × {0})) = 0
 
Theoremi1f1lem 24219* Lemma for i1f1 24220 and itg11 24221. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (𝐹 “ {1}) = 𝐴))
 
Theoremi1f1 24220* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1)
 
Theoremitg11 24221* 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 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫1𝐹) = (vol‘𝐴))
 
Theoremitg1addlem1 24222* Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ⊆ (𝐹 “ {𝑘}))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ dom vol)    &   ((𝜑𝑘𝐴) → (vol‘𝐵) ∈ ℝ)       (𝜑 → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))
 
Theoremi1faddlem 24223* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       ((𝜑𝐴 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))
 
Theoremi1fmullem 24224* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝐴}) = 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))
 
Theoremi1fadd 24225 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
 
Theoremi1fmul 24226 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)
 
Theoremitg1addlem2 24227* Lemma for itg1add 24231. 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 24229 and itg1addlem5 24230. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))       (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
 
Theoremitg1addlem3 24228* Lemma for itg1add 24231. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
 
Theoremitg1addlem4 24229* Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))       (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
 
Theoremitg1addlem5 24230* Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))       (𝜑 → (∫1‘(𝐹f + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
 
Theoremitg1add 24231 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (∫1‘(𝐹f + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
 
Theoremi1fmulclem 24232 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (((𝜑𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → (((ℝ × {𝐴}) ∘f · 𝐹) “ {𝐵}) = (𝐹 “ {(𝐵 / 𝐴)}))
 
Theoremi1fmulc 24233 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → ((ℝ × {𝐴}) ∘f · 𝐹) ∈ dom ∫1)
 
Theoremitg1mulc 24234 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (∫1‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫1𝐹)))
 
Theoremi1fres 24235* 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 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (𝐹𝑥), 0))       ((𝐹 ∈ dom ∫1𝐴 ∈ dom vol) → 𝐺 ∈ dom ∫1)
 
Theoremi1fpos 24236* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))       (𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1)
 
Theoremi1fposd 24237* Deduction form of i1fposd 24237. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1)       (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1)
 
Theoremi1fsub 24238 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1) → (𝐹f𝐺) ∈ dom ∫1)
 
Theoremitg1sub 24239 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1) → (∫1‘(𝐹f𝐺)) = ((∫1𝐹) − (∫1𝐺)))
 
Theoremitg10a 24240* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) = 0)       (𝜑 → (∫1𝐹) = 0)
 
Theoremitg1ge0a 24241* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹𝑥))       (𝜑 → 0 ≤ (∫1𝐹))
 
Theoremitg1lea 24242* Approximate version of itg1le 24243. If 𝐹𝐺 for almost all 𝑥, then 1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   (𝜑𝐺 ∈ dom ∫1)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (∫1𝐹) ≤ (∫1𝐺))
 
Theoremitg1le 24243 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 ∫1𝐺 ∈ dom ∫1𝐹r𝐺) → (∫1𝐹) ≤ (∫1𝐺))
 
Theoremitg1climres 24244* Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐴:ℕ⟶dom vol)    &   ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))    &   (𝜑 ran 𝐴 = ℝ)    &   (𝜑𝐹 ∈ dom ∫1)    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))       (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ (∫1𝐹))
 
Theoremmbfi1fseqlem1 24245* Lemma for mbfi1fseq 24251. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))       (𝜑𝐽:(ℕ × ℝ)⟶(0[,)+∞))
 
Theoremmbfi1fseqlem2 24246* Lemma for mbfi1fseq 24251. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
 
Theoremmbfi1fseqlem3 24247* Lemma for mbfi1fseq 24251. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
 
Theoremmbfi1fseqlem4 24248* Lemma for mbfi1fseq 24251. 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 𝑘 is measurable, because it is equal to (-𝑛[,]𝑛) ∩ (𝐹 “ (𝑘[,)𝑘 + 1 / (2↑𝑛))) for 𝑘 < 𝑛 and (-𝑛[,]𝑛) ∩ (𝐹 “ (𝑘[,)+∞)) for 𝑘 = 𝑛. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝜑𝐺:ℕ⟶dom ∫1)
 
Theoremmbfi1fseqlem5 24249* Lemma for mbfi1fseq 24251. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       ((𝜑𝐴 ∈ ℕ) → (0𝑝r ≤ (𝐺𝐴) ∧ (𝐺𝐴) ∘r ≤ (𝐺‘(𝐴 + 1))))
 
Theoremmbfi1fseqlem6 24250* Lemma for mbfi1fseq 24251. 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
 
Theoremmbfi1fseq 24251* 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
 
Theoremmbfi1flimlem 24252* Lemma for mbfi1flim 24253. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶ℝ)       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
 
Theoremmbfi1flim 24253* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥𝐴 (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
 
Theoremmbfmullem2 24254* Lemma for mbfmul 24256. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   ((𝜑𝑥𝐴) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   (𝜑𝑄:ℕ⟶dom ∫1)    &   ((𝜑𝑥𝐴) → (𝑛 ∈ ℕ ↦ ((𝑄𝑛)‘𝑥)) ⇝ (𝐺𝑥))       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)
 
Theoremmbfmullem 24255 Lemma for mbfmul 24256. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)
 
Theoremmbfmul 24256 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)
 
Theoremitg2lcl 24257* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       𝐿 ⊆ ℝ*
 
Theoremitg2val 24258* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
 
Theoremitg2l 24259* Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))
 
Theoremitg2lr 24260* Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       ((𝐺 ∈ dom ∫1𝐺r𝐹) → (∫1𝐺) ∈ 𝐿)
 
Theoremxrge0f 24261 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:ℝ⟶ℝ ∧ 0𝑝r𝐹) → 𝐹:ℝ⟶(0[,]+∞))
 
Theoremitg2cl 24262 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) ∈ ℝ*)
 
Theoremitg2ub 24263 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:ℝ⟶(0[,]+∞) ∧ 𝐺 ∈ dom ∫1𝐺r𝐹) → (∫1𝐺) ≤ (∫2𝐹))
 
Theoremitg2leub 24264* Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔r𝐹 → (∫1𝑔) ≤ 𝐴)))
 
Theoremitg2ge0 24265 The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2𝐹))
 
Theoremitg2itg1 24266 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 ∧ 0𝑝r𝐹) → (∫2𝐹) = (∫1𝐹))
 
Theoremitg20 24267 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∫2‘(ℝ × {0})) = 0
 
Theoremitg2lecl 24268 If an 2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2𝐹) ≤ 𝐴) → (∫2𝐹) ∈ ℝ)
 
Theoremitg2le 24269 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ 𝐹r𝐺) → (∫2𝐹) ≤ (∫2𝐺))
 
Theoremitg2const 24270* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ (0[,)+∞)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))) = (𝐵 · (vol‘𝐴)))
 
Theoremitg2const2 24271* 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 ∧ 𝐵 ∈ ℝ+) → ((vol‘𝐴) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))) ∈ ℝ))
 
Theoremitg2seq 24272* 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 24285, but unlike that theorem this one doesn't require 𝐹 to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
(𝐹:ℝ⟶(0[,]+∞) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (𝑔𝑛) ∘r𝐹 ∧ (∫2𝐹) = sup(ran (𝑛 ∈ ℕ ↦ (∫1‘(𝑔𝑛))), ℝ*, < )))
 
Theoremitg2uba 24273* Approximate version of itg2ub 24263. If 𝐹 approximately dominates 𝐺, then 1𝐺 ≤ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺𝑥) ≤ (𝐹𝑥))       (𝜑 → (∫1𝐺) ≤ (∫2𝐹))
 
Theoremitg2lea 24274* Approximate version of itg2le 24269. If 𝐹𝐺 for almost all 𝑥, then 2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺:ℝ⟶(0[,]+∞))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (∫2𝐹) ≤ (∫2𝐺))
 
Theoremitg2eqa 24275* Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then 2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺:ℝ⟶(0[,]+∞))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (∫2𝐹) = (∫2𝐺))
 
Theoremitg2mulclem 24276 Lemma for itg2mulc 24277. (Contributed by Mario Carneiro, 8-Jul-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2𝐹)))
 
Theoremitg2mulc 24277 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.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐴 ∈ (0[,)+∞))       (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2𝐹)))
 
Theoremitg2splitlem 24278* Lemma for itg2split 24279. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑𝐵 ∈ dom vol)    &   (𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ (0[,]+∞))    &   𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐶, 0))    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐵, 𝐶, 0))    &   𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥𝑈, 𝐶, 0))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2𝐻) ≤ ((∫2𝐹) + (∫2𝐺)))
 
Theoremitg2split 24279* The 2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 24289 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑𝐵 ∈ dom vol)    &   (𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ (0[,]+∞))    &   𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐶, 0))    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐵, 𝐶, 0))    &   𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥𝑈, 𝐶, 0))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2𝐻) = ((∫2𝐹) + (∫2𝐺)))
 
Theoremitg2monolem1 24280* Lemma for itg2mono 24283. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so 1𝐻𝑆, which is one half of the equality in itg2mono 24283. Consider the sequence 𝐴(𝑛) = {𝑥𝑡 · 𝐻𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is , and so 𝐻𝐴(𝑛) has an integral which equals 1𝐻 in the limit, by itg1climres 24244. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)    &   𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )    &   (𝜑𝑇 ∈ (0(,)1))    &   (𝜑𝐻 ∈ dom ∫1)    &   (𝜑𝐻r𝐺)    &   (𝜑𝑆 ∈ ℝ)    &   𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})       (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)
 
Theoremitg2monolem2 24281* Lemma for itg2mono 24283. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)    &   𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )    &   (𝜑𝑃 ∈ dom ∫1)    &   (𝜑𝑃r𝐺)    &   (𝜑 → ¬ (∫1𝑃) ≤ 𝑆)       (𝜑𝑆 ∈ ℝ)
 
Theoremitg2monolem3 24282* Lemma for itg2mono 24283. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)    &   𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )    &   (𝜑𝑃 ∈ dom ∫1)    &   (𝜑𝑃r𝐺)    &   (𝜑 → ¬ (∫1𝑃) ≤ 𝑆)       (𝜑 → (∫1𝑃) ≤ 𝑆)
 
Theoremitg2mono 24283* The Monotone Convergence Theorem for nonnegative functions. If {(𝐹𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2𝐺) is the limit of the sequence {(∫2‘(𝐹𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)    &   𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )       (𝜑 → (∫2𝐺) = 𝑆)
 
Theoremitg2i1fseqle 24284* Subject to the conditions coming from mbfi1fseq 24251, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))       ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∘r𝐹)
 
Theoremitg2i1fseq 24285* Subject to the conditions coming from mbfi1fseq 24251, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))       (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))
 
Theoremitg2i1fseq2 24286* In an extension to the results of itg2i1fseq 24285, 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑃𝑘)) ≤ 𝑀)       (𝜑𝑆 ⇝ (∫2𝐹))
 
Theoremitg2i1fseq3 24287* Special case of itg2i1fseq2 24286: 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))    &   (𝜑 → (∫2𝐹) ∈ ℝ)       (𝜑𝑆 ⇝ (∫2𝐹))
 
Theoremitg2addlem 24288* Lemma for itg2add 24289. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐺:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐺) ∈ ℝ)    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   (𝜑𝑄:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑄𝑛) ∧ (𝑄𝑛) ∘r ≤ (𝑄‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑄𝑛)‘𝑥)) ⇝ (𝐺𝑥))       (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
 
Theoremitg2add 24289 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐺:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2‘(𝐹f + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))
 
Theoremitg2gt0 24290* 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.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑 → 0 < (vol‘𝐴))    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   ((𝜑𝑥𝐴) → 0 < (𝐹𝑥))       (𝜑 → 0 < (∫2𝐹))
 
Theoremitg2cnlem1 24291* Lemma for itgcn 24372. (Contributed by Mario Carneiro, 30-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)       (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ) = (∫2𝐹))
 
Theoremitg2cnlem2 24292* Lemma for itgcn 24372. (Contributed by Mario Carneiro, 31-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑀, (𝐹𝑥), 0))) ≤ ((∫2𝐹) − (𝐶 / 2)))       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝑢, (𝐹𝑥), 0))) < 𝐶))
 
Theoremitg2cn 24293* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 24563 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝑢, (𝐹𝑥), 0))) < 𝐶))
 
Theoremibllem 24294 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
 
Theoremisibl 24295* The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of 𝐴𝐵 d𝑥, which is to say that the expression 𝐴𝐵 d𝑥 doesn't make sense unless (𝑥𝐴𝐵) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   (𝜑 → dom 𝐹 = 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)       (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
 
Theoremisibl2 24296* 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.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
 
Theoremiblmbf 24297 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
 
Theoremiblitg 24298* 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.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝐾))))    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝐾 ∈ ℤ) → (∫2𝐺) ∈ ℝ)
 
Theoremdfitg 24299* Evaluate the class substitution in df-itg 24153. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))       𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))))
 
Theoremitgex 24300 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐴𝐵 d𝑥 ∈ V
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