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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopnmbllem 24301* Lemma for opnmbl 24302. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Theoremopnmbl 24302 All open sets are measurable. This proof, via dyadmbl 24300 and uniioombl 24289, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

TheoremopnmblALT 24303 All open sets are measurable. This alternative proof of opnmbl 24302 is significantly shorter, at the expense of invoking countable choice ax-cc 9895. (This was also the original proof before the current opnmbl 24302 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Theoremsubopnmbl 24304 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t 𝐴)       ((𝐴 ∈ dom vol ∧ 𝐵𝐽) → 𝐵 ∈ dom vol)

Theoremvolsup2 24305* The volume of 𝐴 is the supremum of the sequence vol*‘(𝐴 ∩ (-𝑛[,]𝑛)) of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))

Theoremvolcn 24306* The function formed by restricting a measurable set to a closed interval with a varying endpoint produces an increasing continuous function on the reals. (Contributed by Mario Carneiro, 30-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ (vol‘(𝐴 ∩ (𝐵[,]𝑥))))       ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ) → 𝐹 ∈ (ℝ–cn→ℝ))

Theoremvolivth 24307* The Intermediate Value Theorem for the Lebesgue volume function. For any positive 𝐵 ≤ (vol‘𝐴), there is a measurable subset of 𝐴 whose volume is 𝐵. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥𝐴 ∧ (vol‘𝑥) = 𝐵))

Theoremvitalilem1 24308* Lemma for vitali 24313. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}        Er (0[,]1)

Theoremvitalilem2 24309* Lemma for vitali 24313. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))

Theoremvitalilem3 24310* Lemma for vitali 24313. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))

Theoremvitalilem4 24311* Lemma for vitali 24313. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))       ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)

Theoremvitalilem5 24312* Lemma for vitali 24313. (Contributed by Mario Carneiro, 16-Jun-2014.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}    &   𝑆 = ((0[,]1) / )    &   (𝜑𝐹 Fn 𝑆)    &   (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))    &   (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))    &   𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})    &   (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))        ¬ 𝜑

Theoremvitali 24313 If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)
( < We ℝ → dom vol ⊊ 𝒫 ℝ)

13.2.2  Lebesgue integration

13.2.2.1  Lesbesgue integral

Syntaxcmbf 24314 Extend class notation with the class of measurable functions.
class MblFn

Syntaxcitg1 24315 Extend class notation with the Lebesgue integral for simple functions.
class 1

Syntaxcitg2 24316 Extend class notation with the Lebesgue integral for nonnegative functions.
class 2

Syntaxcibl 24317 Extend class notation with the class of integrable functions.
class 𝐿1

Syntaxcitg 24318 Extend class notation with the general Lebesgue integral.
class 𝐴𝐵 d𝑥

Definitiondf-mbf 24319* Define the class of measurable functions on the reals. A real function is measurable if the preimage of every open interval is a measurable set (see ismbl 24226) and a complex function is measurable if the real and imaginary parts of the function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}

Definitiondf-itg1 24320* Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)
1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))))

Definitiondf-itg2 24321* Define the Lebesgue integral for nonnegative functions. A nonnegative function's integral is the supremum of the integrals of all simple functions that are less than the input function. Note that this may be +∞ for functions that take the value +∞ on a set of positive measure or functions that are bounded below by a positive number on a set of infinite measure. (Contributed by Mario Carneiro, 28-Jun-2014.)
2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))

Definitiondf-ibl 24322* Define the class of integrable functions on the reals. A function is integrable if it is measurable and the integrals of the pieces of the function are all finite. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}

Definitiondf-itg 24323* Define the full Lebesgue integral, for complex-valued functions to . The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of 𝑥↑2 from 0 to 1 is ∫(0[,]1)(𝑥↑2) d𝑥 = (1 / 3). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 24321 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 24321 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))

Theoremismbf1 24324* The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24328 and ismbfcn 24329 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))

Theoremmbff 24325 A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Theoremmbfdm 24326 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Theoremmbfconstlem 24327 Lemma for mbfconst 24333 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.)
((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → ((𝐴 × {𝐶}) “ 𝐵) ∈ dom vol)

Theoremismbf 24328* The predicate "𝐹 is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 24226. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))

Theoremismbfcn 24329 A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))

Theoremmbfima 24330 Definitional property of a measurable function: the preimage of an open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (𝐹 “ (𝐵(,)𝐶)) ∈ dom vol)

Theoremmbfimaicc 24331 The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐹 “ (𝐵[,]𝐶)) ∈ dom vol)

Theoremmbfimasn 24332 The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ ∧ 𝐵 ∈ ℝ) → (𝐹 “ {𝐵}) ∈ dom vol)

Theoremmbfconst 24333 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn)

Theoremmbf0 24334 The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
∅ ∈ MblFn

Theoremmbfid 24335 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn)

Theoremmbfmptcl 24336* Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)

Theoremmbfdm2 24337* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
(𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑𝐴 ∈ dom vol)

Theoremismbfcn2 24338* A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn ↔ ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)))

Theoremismbfd 24339* Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 24354. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   ((𝜑𝑥 ∈ ℝ*) → (𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    &   ((𝜑𝑥 ∈ ℝ*) → (𝐹 “ (-∞(,)𝑥)) ∈ dom vol)       (𝜑𝐹 ∈ MblFn)

Theoremismbf2d 24340* Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ∈ dom vol)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹 “ (-∞(,)𝑥)) ∈ dom vol)       (𝜑𝐹 ∈ MblFn)

Theoremmbfeqalem1 24341* Lemma for mbfeqalem2 24342. (Contributed by Mario Carneiro, 2-Sep-2014.) (Revised by AV, 19-Aug-2022.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐵) → 𝐷 ∈ ℝ)       (𝜑 → (((𝑥𝐵𝐶) “ 𝑦) ∖ ((𝑥𝐵𝐷) “ 𝑦)) ∈ dom vol)

Theoremmbfeqalem2 24342* Lemma for mbfeqa 24343. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by AV, 19-Aug-2022.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐵) → 𝐷 ∈ ℝ)       (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))

Theoremmbfeqa 24343* If two functions are equal almost everywhere, then one is measurable iff the other is. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐵) → 𝐷 ∈ ℂ)       (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))

Theoremmbfres 24344 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
((𝐹 ∈ MblFn ∧ 𝐴 ∈ dom vol) → (𝐹𝐴) ∈ MblFn)

Theoremmbfres2 24345 Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐵 and 𝐶 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → (𝐹𝐵) ∈ MblFn)    &   (𝜑 → (𝐹𝐶) ∈ MblFn)    &   (𝜑 → (𝐵𝐶) = 𝐴)       (𝜑𝐹 ∈ MblFn)

Theoremmbfss 24346* Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ dom vol)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)       (𝜑 → (𝑥𝐵𝐶) ∈ MblFn)

Theoremmbfmulc2lem 24347 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)

Theoremmbfmulc2re 24348 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:𝐴⟶ℂ)       (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)

Theoremmbfmax 24349* The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺:𝐴⟶ℝ)    &   (𝜑𝐺 ∈ MblFn)    &   𝐻 = (𝑥𝐴 ↦ if((𝐹𝑥) ≤ (𝐺𝑥), (𝐺𝑥), (𝐹𝑥)))       (𝜑𝐻 ∈ MblFn)

Theoremmbfneg 24350* The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)       (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)

Theoremmbfpos 24351* The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)       (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)

Theoremmbfposr 24352* Converse to mbfpos 24351. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)    &   (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)       (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)

Theoremmbfposb 24353* A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))

Theoremismbf3d 24354* Simplified form of ismbfd 24339. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹 “ (𝑥(,)+∞)) ∈ dom vol)       (𝜑𝐹 ∈ MblFn)

Theoremmbfimaopnlem 24355* Lemma for mbfimaopn 24356. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐵 = ((,) “ (ℚ × ℚ))    &   𝐾 = ran (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 × 𝑦))       ((𝐹 ∈ MblFn ∧ 𝐴𝐽) → (𝐹𝐴) ∈ dom vol)

Theoremmbfimaopn 24356 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 24358, which explains why 𝐴 ∈ dom vol is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)       ((𝐹 ∈ MblFn ∧ 𝐴𝐽) → (𝐹𝐴) ∈ dom vol)

Theoremmbfimaopn2 24357 The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐵)       (((𝐹 ∈ MblFn ∧ 𝐹:𝐴𝐵𝐵 ⊆ ℂ) ∧ 𝐶𝐾) → (𝐹𝐶) ∈ dom vol)

Theoremcncombf 24358 The composition of a continuous function with a measurable function is measurable. (More generally, 𝐺 can be a Borel-measurable function, but notably the condition that 𝐺 be only measurable is too weak, the usual counterexample taking 𝐺 to be the Cantor function and 𝐹 the indicator function of the 𝐺-image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ MblFn ∧ 𝐹:𝐴𝐵𝐺 ∈ (𝐵cn→ℂ)) → (𝐺𝐹) ∈ MblFn)

Theoremcnmbf 24359 A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
((𝐴 ∈ dom vol ∧ 𝐹 ∈ (𝐴cn→ℂ)) → 𝐹 ∈ MblFn)

Theoremmbfaddlem 24360 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)       (𝜑 → (𝐹f + 𝐺) ∈ MblFn)

Theoremmbfadd 24361 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹f + 𝐺) ∈ MblFn)

Theoremmbfsub 24362 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹f𝐺) ∈ MblFn)

Theoremmbfmulc2 24363* A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)

Theoremmbfsup 24364* The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, 𝐵(𝑛, 𝑥) is a function of both 𝑛 and 𝑥, since it is an 𝑛-indexed sequence of functions on 𝑥. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = (𝑥𝐴 ↦ sup(ran (𝑛𝑍𝐵), ℝ, < ))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝐵𝑦)       (𝜑𝐺 ∈ MblFn)

Theoremmbfinf 24365* The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)
𝑍 = (ℤ𝑀)    &   𝐺 = (𝑥𝐴 ↦ inf(ran (𝑛𝑍𝐵), ℝ, < ))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑛𝑍 𝑦𝐵)       (𝜑𝐺 ∈ MblFn)

Theoremmbflimsup 24366* The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝑍 = (ℤ𝑀)    &   𝐺 = (𝑥𝐴 ↦ (lim sup‘(𝑛𝑍𝐵)))    &   𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛𝑍𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)       (𝜑𝐺 ∈ MblFn)

Theoremmbflimlem 24367* The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝐴) → (𝑛𝑍𝐵) ⇝ 𝐶)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)       (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)

Theoremmbflim 24368* The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝐴) → (𝑛𝑍𝐵) ⇝ 𝐶)    &   ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵𝑉)       (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)

Syntaxc0p 24369 Extend class notation to include the zero polynomial.
class 0𝑝

Definitiondf-0p 24370 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
0𝑝 = (ℂ × {0})

Theorem0pval 24371 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℂ → (0𝑝𝐴) = 0)

Theorem0plef 24372 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 24373 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 24374 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 24323); 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 24375 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ MblFn)

Theoremi1ff 24376 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)

Theoremi1frn 24377 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)

Theoremi1fima 24378 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (𝐹𝐴) ∈ dom vol)

Theoremi1fima2 24379 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 24380 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(𝐹 “ {𝐴})) ∈ ℝ)

Theoremi1fd 24381* 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 24382 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 24383* 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 24384* 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 24385 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (∫1𝐹) ∈ ℝ)

Theoremitg1ge0 24386 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝r𝐹) → 0 ≤ (∫1𝐹))

Theoremi1f0 24387 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
(ℝ × {0}) ∈ dom ∫1

Theoremitg10 24388 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
(∫1‘(ℝ × {0})) = 0

Theoremi1f1lem 24389* Lemma for i1f1 24390 and itg11 24391. (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 24390* 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 24391* 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 24392* 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 24393* 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 24394* 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 24395 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 24396 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 24397* Lemma for itg1add 24401. 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 24399 and itg1addlem5 24400. (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 24398* Lemma for itg1add 24401. (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 24399* 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 24400* 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)))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45724
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