 Home Metamath Proof ExplorerTheorem List (p. 235 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

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

Theoremuniioombllem5 23401* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)    &   𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))    &   𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))       (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Theoremuniioombllem6 23402* Lemma for uniioombl 23403. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))    &   𝐴 = ran ((,) ∘ 𝐹)    &   (𝜑 → (vol*‘𝐸) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑𝐸 ran ((,) ∘ 𝐺))    &   𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))    &   (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))       (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Theoremuniioombl 23403* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23367.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 ran ((,) ∘ 𝐹) ∈ dom vol)

Theoremuniiccmbl 23404* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23367.) (Contributed by Mario Carneiro, 25-Mar-2015.)
(𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    &   (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))    &   𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))       (𝜑 ran ([,] ∘ 𝐹) ∈ dom vol)

Theoremdyadf 23405* The function 𝐹 returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))

Theoremdyadval 23406* Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)

Theoremdyadovol 23407* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (vol*‘([,]‘(𝐴𝐹𝐵))) = (1 / (2↑𝐵)))

Theoremdyadss 23408* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) → 𝐷𝐶))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) ∧ 𝐶𝐷) → (([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)) ∨ ([,]‘(𝐵𝐹𝐷)) ⊆ ([,]‘(𝐴𝐹𝐶)) ∨ (((,)‘(𝐴𝐹𝐶)) ∩ ((,)‘(𝐵𝐹𝐷))) = ∅))

Theoremdyaddisj 23410* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ∈ ran 𝐹𝐵 ∈ ran 𝐹) → (([,]‘𝐴) ⊆ ([,]‘𝐵) ∨ ([,]‘𝐵) ⊆ ([,]‘𝐴) ∨ (((,)‘𝐴) ∩ ((,)‘𝐵)) = ∅))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑 → ¬ 𝐷 < 𝐶)    &   (𝜑 → ([,]‘(𝐴𝐹𝐶)) ⊆ ([,]‘(𝐵𝐹𝐷)))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Theoremdyadmax 23412* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)       ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))

𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   (𝜑𝐴 ⊆ ran 𝐹)       (𝜑 ([,] “ 𝐴) = ([,] “ 𝐺))

Theoremdyadmbl 23414* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)
𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)    &   𝐺 = {𝑧𝐴 ∣ ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}    &   (𝜑𝐴 ⊆ ran 𝐹)       (𝜑 ([,] “ 𝐴) ∈ dom vol)

Theoremopnmbllem 23415* Lemma for opnmbl 23416. (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 23416 All open sets are measurable. This proof, via dyadmbl 23414 and uniioombl 23403, 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 23417 All open sets are measurable. This alternative proof of opnmbl 23416 is significantly shorter, at the expense of invoking countable choice ax-cc 9295. (This was also the original proof before the current opnmbl 23416 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Theoremsubopnmbl 23418 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 23419* 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 23420* 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 23421* 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 23422* Lemma for vitali 23427. (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 23423* Lemma for vitali 23427. (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 23424* Lemma for vitali 23427. (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 23425* Lemma for vitali 23427. (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 23426* Lemma for vitali 23427. (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 23427 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 23428 Extend class notation with the class of measurable functions.
class MblFn

Syntaxcitg1 23429 Extend class notation with the Lebesgue integral for simple functions.
class 1

Syntaxcitg2 23430 Extend class notation with the Lebesgue integral for nonnegative functions.
class 2

Syntaxcibl 23431 Extend class notation with the class of integrable functions.
class 𝐿1

Syntaxcitg 23432 Extend class notation with the general Lebesgue integral.
class 𝐴𝐵 d𝑥

Definitiondf-mbf 23433* 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 23340) 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 23434* 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 23435* 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 23436* 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 23437* 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 23435 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 23435 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 23438* The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 23442 and ismbfcn 23443 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 23439 A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Theoremmbfdm 23440 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)

Theoremmbfconstlem 23441 Lemma for mbfconst 23447. (Contributed by Mario Carneiro, 17-Jun-2014.)
((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → ((𝐴 × {𝐶}) “ 𝐵) ∈ dom vol)

Theoremismbf 23442* 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 23340. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))

Theoremismbfcn 23443 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 23444 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 23445 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 23446 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 23447 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn)

Theoremmbfid 23448 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
(𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn)

Theoremmbfmptcl 23449* 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 23450* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
(𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑𝐴 ∈ dom vol)

Theoremismbfcn2 23451* 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 23452* 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 23466. (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 23453* 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 23454* Lemma for mbfeqa 23455. (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 23455* 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 23456 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
((𝐹 ∈ MblFn ∧ 𝐴 ∈ dom vol) → (𝐹𝐴) ∈ MblFn)

Theoremmbfres2 23457 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 23458* 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 23459 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 23460 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 23461* 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 23462* 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 23463* 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 23464* Converse to mbfpos 23463. (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 23465* 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 23466* Simplified form of ismbfd 23452. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹:𝐴⟶ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹 “ (𝑥(,)+∞)) ∈ dom vol)       (𝜑𝐹 ∈ MblFn)

Theoremmbfimaopnlem 23467* Lemma for mbfimaopn 23468. (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 23468 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 23470, 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 23469 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 23470 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 23471 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 23472 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 23473 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓 + 𝐺) ∈ MblFn)

Theoremmbfsub 23474 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓𝐺) ∈ MblFn)

Theoremmbfmulc2 23475* 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 23476* 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 23477* 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 23478* 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 23479* 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 23480* 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 23481 Extend class notation to include the zero polynomial.
class 0𝑝

Definitiondf-0p 23482 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
0𝑝 = (ℂ × {0})

Theorem0pval 23483 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℂ → (0𝑝𝐴) = 0)

Theorem0plef 23484 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 23485 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 23486 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 23437); 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 23487 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ MblFn)

Theoremi1ff 23488 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)

Theoremi1frn 23489 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)

Theoremi1fima 23490 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (𝐹𝐴) ∈ dom vol)

Theoremi1fima2 23491 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 23492 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
((𝐹 ∈ dom ∫1𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(𝐹 “ {𝐴})) ∈ ℝ)

Theoremi1fd 23493* 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 23494 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 23495* 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 23496* 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 23497 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐹 ∈ dom ∫1 → (∫1𝐹) ∈ ℝ)

Theoremitg1ge0 23498 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝𝑟𝐹) → 0 ≤ (∫1𝐹))

Theoremi1f0 23499 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
(ℝ × {0}) ∈ dom ∫1

Theoremitg10 23500 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
(∫1‘(ℝ × {0})) = 0

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >