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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremismbfd 23401* 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 23415. (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 23402* 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)

Theoremmbfeqalem 23403* Lemma for mbfeqa 23404. (Contributed by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐵) → 𝐷 ∈ ℝ)       (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))

Theoremmbfeqa 23404* 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 23405 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
((𝐹 ∈ MblFn ∧ 𝐴 ∈ dom vol) → (𝐹𝐴) ∈ MblFn)

Theoremmbfres2 23406 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 23407* 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 23408 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 23409 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 23410* 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 23411* 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 23412* 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 23413* Converse to mbfpos 23412. (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 23414* 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 23415* Simplified form of ismbfd 23401. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹 “ (𝑥(,)+∞)) ∈ dom vol)       (𝜑𝐹 ∈ MblFn)

Theoremmbfimaopnlem 23416* Lemma for mbfimaopn 23417. (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 23417 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 23419, 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 23418 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 23419 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 23420 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 23421 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 23422 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓 + 𝐺) ∈ MblFn)

Theoremmbfsub 23423 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓𝐺) ∈ MblFn)

Theoremmbfmulc2 23424* 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 23425* 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 23426* 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 23427* 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 23428* 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 23429* 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 23430 Extend class notation to include the zero polynomial.
class 0𝑝

Definitiondf-0p 23431 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
0𝑝 = (ℂ × {0})

Theorem0pval 23432 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℂ → (0𝑝𝐴) = 0)

Theorem0plef 23433 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 23434 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 23435 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 23386); 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 23436 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ MblFn)

Theoremi1ff 23437 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)

Theoremi1frn 23438 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)

Theoremi1fima 23439 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (𝐹𝐴) ∈ dom vol)

Theoremi1fima2 23440 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 23441 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(𝐹 “ {𝐴})) ∈ ℝ)

Theoremi1fd 23442* 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 23443 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 23444* 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 23445* 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 23446 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (∫1𝐹) ∈ ℝ)

Theoremitg1ge0 23447 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝𝑟𝐹) → 0 ≤ (∫1𝐹))

Theoremi1f0 23448 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
(ℝ × {0}) ∈ dom ∫1

Theoremitg10 23449 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
(∫1‘(ℝ × {0})) = 0

Theoremi1f1lem 23450* Lemma for i1f1 23451 and itg11 23452. (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 23451* 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 23452* 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 23453* 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 23454* 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 23455* 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 23456 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 23457 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 23458* Lemma for itg1add 23462. 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 23460 and itg1addlem5 23461. (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)

(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}))))

(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)))

(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 23462 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 23463 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 23464 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 23465 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 23466* 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 23467* 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 23468* Deduction form of i1fposd 23468. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1)       (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1)

Theoremi1fsub 23469 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 23470 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 23471* 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 23472* 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 23473* Approximate version of itg1le 23474. 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 23474 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 23475* 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 23476* Lemma for mbfi1fseq 23482. (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 23477* Lemma for mbfi1fseq 23482. (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 23478* Lemma for mbfi1fseq 23482. (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 23479* Lemma for mbfi1fseq 23482. 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 23480* Lemma for mbfi1fseq 23482. 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 23481* Lemma for mbfi1fseq 23482. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existence 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 23482* 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 23483* Lemma for mbfi1flim 23484. (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 23484* 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 23485* Lemma for mbfmul 23487. (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 23486 Lemma for mbfmul 23487. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ MblFn)

Theoremmbfmul 23487 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ MblFn)

Theoremitg2lcl 23488* 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 23489* 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 23490* 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 23491* 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 23492 A real function is a nonnegative extended real function if all its values are greater 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 23493 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 23494 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 23495* 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 23496 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2𝐹))

Theoremitg2itg1 23497 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 23498 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∫2‘(ℝ × {0})) = 0

Theoremitg2lecl 23499 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 23500 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))

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