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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dyadmax 24201* | 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 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)) | ||
Theorem | dyadmbllem 24202* | Lemma for dyadmbl 24203. (Contributed by Mario Carneiro, 26-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) & ⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} & ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) ⇒ ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺)) | ||
Theorem | dyadmbl 24203* | 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 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} & ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) ⇒ ⊢ (𝜑 → ∪ ([,] “ 𝐴) ∈ dom vol) | ||
Theorem | opnmbllem 24204* | Lemma for opnmbl 24205. (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) | ||
Theorem | opnmbl 24205 | All open sets are measurable. This proof, via dyadmbl 24203 and uniioombl 24192, 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) | ||
Theorem | opnmblALT 24206 | All open sets are measurable. This alternative proof of opnmbl 24205 is significantly shorter, at the expense of invoking countable choice ax-cc 9859. (This was also the original proof before the current opnmbl 24205 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol) | ||
Theorem | subopnmbl 24207 | 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 ∧ 𝐵 ∈ 𝐽) → 𝐵 ∈ dom vol) | ||
Theorem | volsup2 24208* | 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)))) | ||
Theorem | volcn 24209* | 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)) | ||
Theorem | volivth 24210* | 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(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵)) | ||
Theorem | vitalilem1 24211* | Lemma for vitali 24216. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} ⇒ ⊢ ∼ Er (0[,]1) | ||
Theorem | vitalilem2 24212* | Lemma for vitali 24216. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} & ⊢ 𝑆 = ((0[,]1) / ∼ ) & ⊢ (𝜑 → 𝐹 Fn 𝑆) & ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) & ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹}) & ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)) ⇒ ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))) | ||
Theorem | vitalilem3 24213* | Lemma for vitali 24216. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} & ⊢ 𝑆 = ((0[,]1) / ∼ ) & ⊢ (𝜑 → 𝐹 Fn 𝑆) & ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) & ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹}) & ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)) ⇒ ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚)) | ||
Theorem | vitalilem4 24214* | Lemma for vitali 24216. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} & ⊢ 𝑆 = ((0[,]1) / ∼ ) & ⊢ (𝜑 → 𝐹 Fn 𝑆) & ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) & ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹}) & ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)) ⇒ ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0) | ||
Theorem | vitalilem5 24215* | Lemma for vitali 24216. (Contributed by Mario Carneiro, 16-Jun-2014.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)} & ⊢ 𝑆 = ((0[,]1) / ∼ ) & ⊢ (𝜑 → 𝐹 Fn 𝑆) & ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)) & ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) & ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹}) & ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | vitali 24216 | 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) | ||
Syntax | cmbf 24217 | Extend class notation with the class of measurable functions. |
class MblFn | ||
Syntax | citg1 24218 | Extend class notation with the Lebesgue integral for simple functions. |
class ∫1 | ||
Syntax | citg2 24219 | Extend class notation with the Lebesgue integral for nonnegative functions. |
class ∫2 | ||
Syntax | cibl 24220 | Extend class notation with the class of integrable functions. |
class 𝐿1 | ||
Syntax | citg 24221 | Extend class notation with the general Lebesgue integral. |
class ∫𝐴𝐵 d𝑥 | ||
Definition | df-mbf 24222* | 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 24129) 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)} | ||
Definition | df-itg1 24223* | 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})))) | ||
Definition | df-itg2 24224* | 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(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) | ||
Definition | df-ibl 24225* | 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‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ} | ||
Definition | df-itg 24226* | Define the full Lebesgue integral, for complex-valued functions to ℝ. 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 24224 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 24224 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))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) | ||
Theorem | ismbf1 24227* | The predicate "𝐹 is a measurable function". This is more naturally stated for functions on the reals, see ismbf 24231 and ismbfcn 24232 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))) | ||
Theorem | mbff 24228 | A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | ||
Theorem | mbfdm 24229 | The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | ||
Theorem | mbfconstlem 24230 | Lemma for mbfconst 24236 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) | ||
Theorem | ismbf 24231* | 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 24129. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | ||
Theorem | ismbfcn 24232 | 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))) | ||
Theorem | mbfima 24233 | 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) | ||
Theorem | mbfimaicc 24234 | 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) | ||
Theorem | mbfimasn 24235 | 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) | ||
Theorem | mbfconst 24236 | A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) | ||
Theorem | mbf0 24237 | The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.) |
⊢ ∅ ∈ MblFn | ||
Theorem | mbfid 24238 | The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) | ||
Theorem | mbfmptcl 24239* | Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | ||
Theorem | mbfdm2 24240* | The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ dom vol) | ||
Theorem | ismbfcn2 24241* | A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) | ||
Theorem | ismbfd 24242* | 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 24257. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | ismbf2d 24243* | Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfeqalem1 24244* | Lemma for mbfeqalem2 24245. (Contributed by Mario Carneiro, 2-Sep-2014.) (Revised by AV, 19-Aug-2022.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℝ) ⇒ ⊢ (𝜑 → ((◡(𝑥 ∈ 𝐵 ↦ 𝐶) “ 𝑦) ∖ (◡(𝑥 ∈ 𝐵 ↦ 𝐷) “ 𝑦)) ∈ dom vol) | ||
Theorem | mbfeqalem2 24245* | Lemma for mbfeqa 24246. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by AV, 19-Aug-2022.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn)) | ||
Theorem | mbfeqa 24246* | 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.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn)) | ||
Theorem | mbfres 24247 | The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ dom vol) → (𝐹 ↾ 𝐴) ∈ MblFn) | ||
Theorem | mbfres2 24248 | 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.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) & ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) & ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfss 24249* | Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn) | ||
Theorem | mbfmulc2lem 24250 | Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn) | ||
Theorem | mbfmulc2re 24251 | Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn) | ||
Theorem | mbfmax 24252* | The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥))) ⇒ ⊢ (𝜑 → 𝐻 ∈ MblFn) | ||
Theorem | mbfneg 24253* | The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn) | ||
Theorem | mbfpos 24254* | The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) | ||
Theorem | mbfposr 24255* | Converse to mbfpos 24254. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | ||
Theorem | mbfposb 24256* | A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))) | ||
Theorem | ismbf3d 24257* | Simplified form of ismbfd 24242. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) ⇒ ⊢ (𝜑 → 𝐹 ∈ MblFn) | ||
Theorem | mbfimaopnlem 24258* | Lemma for mbfimaopn 24259. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) & ⊢ 𝐵 = ((,) “ (ℚ × ℚ)) & ⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol) | ||
Theorem | mbfimaopn 24259 | The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 24261, 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 ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol) | ||
Theorem | mbfimaopn2 24260 | 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 = (𝐽 ↾t 𝐵) ⇒ ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝐶 ∈ 𝐾) → (◡𝐹 “ 𝐶) ∈ dom vol) | ||
Theorem | cncombf 24261 | 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:𝐴⟶𝐵 ∧ 𝐺 ∈ (𝐵–cn→ℂ)) → (𝐺 ∘ 𝐹) ∈ MblFn) | ||
Theorem | cnmbf 24262 | A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.) |
⊢ ((𝐴 ∈ dom vol ∧ 𝐹 ∈ (𝐴–cn→ℂ)) → 𝐹 ∈ MblFn) | ||
Theorem | mbfaddlem 24263 | The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn) | ||
Theorem | mbfadd 24264 | The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn) | ||
Theorem | mbfsub 24265 | The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) ⇒ ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ MblFn) | ||
Theorem | mbfmulc2 24266* | A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) | ||
Theorem | mbfsup 24267* | 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 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → 𝐺 ∈ MblFn) | ||
Theorem | mbfinf 24268* | 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 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐺 ∈ MblFn) | ||
Theorem | mbflimsup 24269* | 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 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) & ⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < )) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐺 ∈ MblFn) | ||
Theorem | mbflimlem 24270* | The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) | ||
Theorem | mbflim 24271* | The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) | ||
Syntax | c0p 24272 | Extend class notation to include the zero polynomial. |
class 0𝑝 | ||
Definition | df-0p 24273 | Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ 0𝑝 = (ℂ × {0}) | ||
Theorem | 0pval 24274 | The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | ||
Theorem | 0plef 24275 | 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 ∧ 0𝑝 ∘r ≤ 𝐹)) | ||
Theorem | 0pledm 24276 | Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹 Fn 𝐴) ⇒ ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) | ||
Theorem | isi1f 24277 | 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 24226); 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))) | ||
Theorem | i1fmbf 24278 | Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | ||
Theorem | i1ff 24279 | A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | ||
Theorem | i1frn 24280 | A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | ||
Theorem | i1fima 24281 | Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) | ||
Theorem | i1fima2 24282 | 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‘(◡𝐹 “ 𝐴)) ∈ ℝ) | ||
Theorem | i1fima2sn 24283 | Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(◡𝐹 “ {𝐴})) ∈ ℝ) | ||
Theorem | i1fd 24284* | A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ran 𝐹 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ∫1) | ||
Theorem | i1f0rn 24285 | 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) | ||
Theorem | itg1val 24286* | The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | ||
Theorem | itg1val2 24287* | The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ 𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) → (∫1‘𝐹) = Σ𝑥 ∈ 𝐴 (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | ||
Theorem | itg1cl 24288 | Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | ||
Theorem | itg1ge0 24289 | Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝐹) → 0 ≤ (∫1‘𝐹)) | ||
Theorem | i1f0 24290 | The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (ℝ × {0}) ∈ dom ∫1 | ||
Theorem | itg10 24291 | The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (∫1‘(ℝ × {0})) = 0 | ||
Theorem | i1f1lem 24292* | Lemma for i1f1 24293 and itg11 24294. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) | ||
Theorem | i1f1 24293* | Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) | ||
Theorem | itg11 24294* | 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(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫1‘𝐹) = (vol‘𝐴)) | ||
Theorem | itg1addlem1 24295* | Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ (◡𝐹 “ {𝑘})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)) | ||
Theorem | i1faddlem 24296* | Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) | ||
Theorem | i1fmullem 24297* | Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) | ||
Theorem | i1fadd 24298 | The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1) | ||
Theorem | i1fmul 24299 | The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1) | ||
Theorem | itg1addlem2 24300* | Lemma for itg1add 24304. 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 24302 and itg1addlem5 24303. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) ⇒ ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ) |
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