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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | truae 31401* | A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ∪ dom 𝑀 = 𝑂 & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) | ||
Theorem | aean 31402* | A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
⊢ ∪ dom 𝑀 = 𝑂 ⇒ ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀))) | ||
Definition | df-fae 31403* | Define a builder for an 'almost everywhere' relation between functions, from relations between function values. In this definition, the range of 𝑓 and 𝑔 is enforced in order to ensure the resulting relation is a set. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ~ a.e. = (𝑟 ∈ V, 𝑚 ∈ ∪ ran measures ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)}) | ||
Theorem | faeval 31404* | Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}) | ||
Theorem | relfae 31405 | The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀)) | ||
Theorem | brfae 31406* | 'almost everywhere' relation for two functions 𝐹 and 𝐺 with regard to the measure 𝑀. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ dom 𝑅 = 𝐷 & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀)) & ⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀)) ⇒ ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)) | ||
Syntax | cmbfm 31407 | Extend class notation with the measurable functions builder. |
class MblFnM | ||
Definition | df-mbfm 31408* |
Define the measurable function builder, which generates the set of
measurable functions from a measurable space to another one. Here, the
measurable spaces are given using their sigma-algebras 𝑠 and
𝑡,
and the spaces themselves are recovered by ∪ 𝑠 and ∪ 𝑡.
Note the similarities between the definition of measurable functions in measure theory, and of continuous functions in topology. This is the definition for the generic measure theory. For the specific case of functions from ℝ to ℂ, see df-mbf 24147. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) | ||
Theorem | ismbfm 31409* | The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24156. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) | ||
Theorem | elunirnmbfm 31410* | The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) | ||
Theorem | mbfmfun 31411 | A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | mbfmf 31412 | A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) | ||
Theorem | isanmbfm 31413 | The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) | ||
Theorem | mbfmcnvima 31414 | The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) ⇒ ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) | ||
Theorem | mbfmbfm 31415 | A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) | ||
Theorem | mbfmcst 31416* | A constant function is measurable. Cf. mbfconst 24161. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑇) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | ||
Theorem | 1stmbfm 31417 | The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (1st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆)) | ||
Theorem | 2ndmbfm 31418 | The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) ⇒ ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇)) | ||
Theorem | imambfm 31419* | If the sigma-algebra in the range of a given function is generated by a collection of basic sets 𝐾, then to check the measurability of that function, we need only consider inverse images of basic sets 𝑎. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
⊢ (𝜑 → 𝐾 ∈ V) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))) | ||
Theorem | cnmbfm 31420 | A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑆 = (sigaGen‘𝐽)) & ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | ||
Theorem | mbfmco 31421 | The composition of two measurable functions is measurable. ( cf. cnmpt11 22199) (Contributed by Thierry Arnoux, 4-Jun-2017.) |
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇)) | ||
Theorem | mbfmco2 31422* | The pair building of two measurable functions is measurable. ( cf. cnmpt1t 22201). (Contributed by Thierry Arnoux, 6-Jun-2017.) |
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) & ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇)) & ⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) ⇒ ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇))) | ||
Theorem | mbfmvolf 31423 | Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.) |
⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ) | ||
Theorem | elmbfmvol2 31424 | Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) | ||
Theorem | mbfmcnt 31425 | All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂)) | ||
Theorem | br2base 31426* | The base set for the generator of the Borel sigma-algebra on (ℝ × ℝ) is indeed (ℝ × ℝ). (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ) | ||
Theorem | dya2ub 31427 | An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ (𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅) | ||
Theorem | sxbrsigalem0 31428* | The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.) |
⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ) | ||
Theorem | sxbrsigalem3 31429* | The sigma-algebra generated by the closed half-spaces of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed sets of (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ⊆ (sigaGen‘(Clsd‘(𝐽 ×t 𝐽))) | ||
Theorem | dya2iocival 31430* | The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 24128. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) | ||
Theorem | dya2iocress 31431* | Dyadic intervals are subsets of ℝ. (Contributed by Thierry Arnoux, 18-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ⊆ ℝ) | ||
Theorem | dya2iocbrsiga 31432* | Dyadic intervals are Borel sets of ℝ. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅ℝ) | ||
Theorem | dya2icobrsiga 31433* | Dyadic intervals are Borel sets of ℝ. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ⇒ ⊢ ran 𝐼 ⊆ 𝔅ℝ | ||
Theorem | dya2icoseg 31434* | For any point and any closed-below, open-above interval of ℝ centered on that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑁 = (⌊‘(1 − (2 logb 𝐷))) ⇒ ⊢ ((𝑋 ∈ ℝ ∧ 𝐷 ∈ ℝ+) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝐷)(,)(𝑋 + 𝐷)))) | ||
Theorem | dya2icoseg2 31435* | For any point and any open interval of ℝ containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ⇒ ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) | ||
Theorem | dya2iocrfn 31436* | The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) | ||
Theorem | dya2iocct 31437* | The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ ran 𝑅 ≼ ω | ||
Theorem | dya2iocnrect 31438* | For any point of an open rectangle in (ℝ × ℝ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) & ⊢ 𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) | ||
Theorem | dya2iocnei 31439* | For any point of an open set of the usual topology on (ℝ × ℝ) there is a closed-below open-above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) | ||
Theorem | dya2iocuni 31440* | Every open set of (ℝ × ℝ) is a union of closed-below open-above dyadic rational rectangular subsets of (ℝ × ℝ). This union must be a countable union by dya2iocct 31437. (Contributed by Thierry Arnoux, 18-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ (𝐴 ∈ (𝐽 ×t 𝐽) → ∃𝑐 ∈ 𝒫 ran 𝑅∪ 𝑐 = 𝐴) | ||
Theorem | dya2iocucvr 31441* | The dyadic rectangular set collection covers (ℝ × ℝ). (Contributed by Thierry Arnoux, 18-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ ∪ ran 𝑅 = (ℝ × ℝ) | ||
Theorem | sxbrsigalem1 31442* | The Borel algebra on (ℝ × ℝ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅) | ||
Theorem | sxbrsigalem2 31443* | The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed half-spaces of (ℝ × ℝ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) | ||
Theorem | sxbrsigalem4 31444* | The Borel algebra on (ℝ × ℝ) is generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed-below, open-above instead of open-below, closed-above in the proof as they are ultimately generated by the floor function. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅) | ||
Theorem | sxbrsigalem5 31445* | First direction for sxbrsiga 31447. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) & ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) ⇒ ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) | ||
Theorem | sxbrsigalem6 31446 | First direction for sxbrsiga 31447, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) | ||
Theorem | sxbrsiga 31447 | The product sigma-algebra (𝔅ℝ ×s 𝔅ℝ) is the Borel algebra on (ℝ × ℝ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.) |
⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘(𝐽 ×t 𝐽)) | ||
In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure 𝑅. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 31454) - it is monotone (omsmon 31455) - it is countably sub-additive (omssubadd 31457) See Definition 1.11.1 of [Bogachev] p. 41. | ||
Syntax | coms 31448 | Class declaration for the outer measure construction function. |
class toOMeas | ||
Definition | df-oms 31449* | Define a function constructing an outer measure. See omsval 31450 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < ))) | ||
Theorem | omsval 31450* | Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))) | ||
Theorem | omsfval 31451* | Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )) | ||
Theorem | omscl 31452* | A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞)) | ||
Theorem | omsf 31453 | A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 ∪ dom 𝑅⟶(0[,]+∞)) | ||
Theorem | oms0 31454 | A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → ∅ ∈ dom 𝑅) & ⊢ (𝜑 → (𝑅‘∅) = 0) ⇒ ⊢ (𝜑 → (𝑀‘∅) = 0) | ||
Theorem | omsmon 31455 | A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑄) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | omssubaddlem 31456* | For any small margin 𝐸, we can find a covering approaching the outer measure of a set 𝐴 by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑤 ∈ 𝑥(𝑅‘𝑤) < ((𝑀‘𝐴) + 𝐸)) | ||
Theorem | omssubadd 31457* | A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → 𝑋 ≼ ω) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴)) | ||
Syntax | ccarsg 31458 | Class declaration for the Caratheodory sigma-Algebra construction. |
class toCaraSiga | ||
Definition | df-carsg 31459* | Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 31460 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)}) | ||
Theorem | carsgval 31460* | Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) | ||
Theorem | carsgcl 31461 | Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) | ||
Theorem | elcarsg 31462* | Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) | ||
Theorem | baselcarsg 31463 | The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | 0elcarsg 31464 | The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsguni 31465 | The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) | ||
Theorem | elcarsgss 31466 | Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑂) | ||
Theorem | difelcarsg 31467 | The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | inelcarsg 31468* | The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | unelcarsg 31469* | The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | difelcarsg2 31470* | The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgmon 31471* | Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
Theorem | carsgsigalem 31472* | Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) | ||
Theorem | fiunelcarsg 31473* | The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgclctunlem1 31474* | Lemma for carsgclctun 31478. (Contributed by Thierry Arnoux, 23-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦))) | ||
Theorem | carsggect 31475* | The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → ¬ ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴)) | ||
Theorem | carsgclctunlem2 31476* | Lemma for carsgclctun 31478. (Contributed by Thierry Arnoux, 25-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) & ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸)) | ||
Theorem | carsgclctunlem3 31477* | Lemma for carsgclctun 31478. (Contributed by Thierry Arnoux, 24-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸)) | ||
Theorem | carsgclctun 31478* | The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
Theorem | carsgsiga 31479* | The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂)) | ||
Theorem | omsmeas 31480 | The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.) |
⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ 𝑆 = (toCaraSiga‘𝑀) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → ∅ ∈ dom 𝑅) & ⊢ (𝜑 → (𝑅‘∅) = 0) ⇒ ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆)) | ||
Theorem | pmeasmono 31481* | This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵)) | ||
Theorem | pmeasadd 31482* | A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} & ⊢ (𝜑 → 𝑅 ∈ 𝑄) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) | ||
Theorem | itgeq12dv 31483* | Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥) | ||
Syntax | citgm 31484 | Extend class notation with the (measure) Bochner integral. |
class itgm | ||
Syntax | csitm 31485 | Extend class notation with the integral metric for simple functions. |
class sitm | ||
Syntax | csitg 31486 | Extend class notation with the integral of simple functions. |
class sitg | ||
Definition | df-sitg 31487* |
Define the integral of simple functions from a measurable space
dom 𝑚 to a generic space 𝑤
equipped with the right scalar
product. 𝑤 will later be required to be a Banach
space.
These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition. Moreover, for each 𝑥, the pre-image (◡𝑔 “ {𝑥}) is requested to be measurable, of finite measure. In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra. Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.) |
⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥))))) | ||
Definition | df-sitm 31488* | Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔)))) | ||
Theorem | sitgval 31489* | Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) ⇒ ⊢ (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥))))) | ||
Theorem | issibf 31490* | The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))) | ||
Theorem | sibf0 31491 | The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ Mnd) ⇒ ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀)) | ||
Theorem | sibfmbl 31492 | A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) | ||
Theorem | sibff 31493 | A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) | ||
Theorem | sibfrn 31494 | A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ Fin) | ||
Theorem | sibfima 31495 | Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)) | ||
Theorem | sibfinima 31496 | The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝐽 ∈ Fre) ⇒ ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞)) | ||
Theorem | sibfof 31497 | Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ 𝐶 = (Base‘𝐾) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐾 ∈ TopSp) & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾)) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀)) | ||
Theorem | sitgfval 31498* | Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥)))) | ||
Theorem | sitgclg 31499* | Closure of the Bochner integral on simple functions, generic version. See sitgclbn 31500 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) & ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) | ||
Theorem | sitgclbn 31500 | Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ Ban) & ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) |
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