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Type | Label | Description |
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Statement | ||
Theorem | faeval 33801* | Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ ((π β V β§ π β βͺ ran measures) β (π ~ a.e.π) = {β¨π, πβ© β£ ((π β (dom π βm βͺ dom π) β§ π β (dom π βm βͺ dom π)) β§ {π₯ β βͺ dom π β£ (πβπ₯)π (πβπ₯)}a.e.π)}) | ||
Theorem | relfae 33802 | The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ ((π β V β§ π β βͺ ran measures) β Rel (π ~ a.e.π)) | ||
Theorem | brfae 33803* | 'almost everywhere' relation for two functions πΉ and πΊ with regard to the measure π. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ dom π = π· & β’ (π β π β V) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β (π· βm βͺ dom π)) & β’ (π β πΊ β (π· βm βͺ dom π)) β β’ (π β (πΉ(π ~ a.e.π)πΊ β {π₯ β βͺ dom π β£ (πΉβπ₯)π (πΊβπ₯)}a.e.π)) | ||
Syntax | cmbfm 33804 | Extend class notation with the measurable functions builder. |
class MblFnM | ||
Definition | df-mbfm 33805* |
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 π and
π‘,
and the spaces themselves are recovered by βͺ π and βͺ π‘.
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 β to β, see df-mbf 25535. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
β’ MblFnM = (π β βͺ ran sigAlgebra, π‘ β βͺ ran sigAlgebra β¦ {π β (βͺ π‘ βm βͺ π ) β£ βπ₯ β π‘ (β‘π β π₯) β π }) | ||
Theorem | ismbfm 33806* | The predicate "πΉ is a measurable function from the measurable space π to the measurable space π". Cf. ismbf 25544. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) β β’ (π β (πΉ β (πMblFnMπ) β (πΉ β (βͺ π βm βͺ π) β§ βπ₯ β π (β‘πΉ β π₯) β π))) | ||
Theorem | elunirnmbfm 33807* | The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
β’ (πΉ β βͺ ran MblFnM β βπ β βͺ ran sigAlgebraβπ‘ β βͺ ran sigAlgebra(πΉ β (βͺ π‘ βm βͺ π ) β§ βπ₯ β π‘ (β‘πΉ β π₯) β π )) | ||
Theorem | mbfmfun 33808 | A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
β’ (π β πΉ β βͺ ran MblFnM) β β’ (π β Fun πΉ) | ||
Theorem | mbfmf 33809 | A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ β (πMblFnMπ)) β β’ (π β πΉ:βͺ πβΆβͺ π) | ||
Theorem | isanmbfmOLD 33810 | Obsolete version of isanmbfm 33812 as of 13-Jan-2025. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ β (πMblFnMπ)) β β’ (π β πΉ β βͺ ran MblFnM) | ||
Theorem | mbfmcnvima 33811 | The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ β (πMblFnMπ)) & β’ (π β π΄ β π) β β’ (π β (β‘πΉ β π΄) β π) | ||
Theorem | isanmbfm 33812 | The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.) |
β’ (π β πΉ β (πMblFnMπ)) β β’ (π β πΉ β βͺ ran MblFnM) | ||
Theorem | mbfmbfmOLD 33813 | A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π β βͺ ran measures) & β’ (π β π½ β Top) & β’ (π β πΉ β (dom πMblFnM(sigaGenβπ½))) β β’ (π β πΉ β βͺ ran MblFnM) | ||
Theorem | mbfmbfm 33814 | A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.) |
β’ (π β πΉ β (dom πMblFnM(sigaGenβπ½))) β β’ (π β πΉ β βͺ ran MblFnM) | ||
Theorem | mbfmcst 33815* | A constant function is measurable. Cf. mbfconst 25549. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ = (π₯ β βͺ π β¦ π΄)) & β’ (π β π΄ β βͺ π) β β’ (π β πΉ β (πMblFnMπ)) | ||
Theorem | 1stmbfm 33816 | The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) β β’ (π β (1st βΎ (βͺ π Γ βͺ π)) β ((π Γs π)MblFnMπ)) | ||
Theorem | 2ndmbfm 33817 | The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) β β’ (π β (2nd βΎ (βͺ π Γ βͺ π)) β ((π Γs π)MblFnMπ)) | ||
Theorem | imambfm 33818* | If the sigma-algebra in the range of a given function is generated by a collection of basic sets πΎ, then to check the measurability of that function, we need only consider inverse images of basic sets π. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
β’ (π β πΎ β V) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π = (sigaGenβπΎ)) β β’ (π β (πΉ β (πMblFnMπ) β (πΉ:βͺ πβΆβͺ π β§ βπ β πΎ (β‘πΉ β π) β π))) | ||
Theorem | cnmbfm 33819 | A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
β’ (π β πΉ β (π½ Cn πΎ)) & β’ (π β π = (sigaGenβπ½)) & β’ (π β π = (sigaGenβπΎ)) β β’ (π β πΉ β (πMblFnMπ)) | ||
Theorem | mbfmco 33820 | The composition of two measurable functions is measurable. See cnmpt11 23554. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ β (π MblFnMπ)) & β’ (π β πΊ β (πMblFnMπ)) β β’ (π β (πΊ β πΉ) β (π MblFnMπ)) | ||
Theorem | mbfmco2 33821* | The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23556). (Contributed by Thierry Arnoux, 6-Jun-2017.) |
β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β π β βͺ ran sigAlgebra) & β’ (π β πΉ β (π MblFnMπ)) & β’ (π β πΊ β (π MblFnMπ)) & β’ π» = (π₯ β βͺ π β¦ β¨(πΉβπ₯), (πΊβπ₯)β©) β β’ (π β π» β (π MblFnM(π Γs π))) | ||
Theorem | mbfmvolf 33822 | Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.) |
β’ (πΉ β (dom volMblFnMπ β) β πΉ:ββΆβ) | ||
Theorem | elmbfmvol2 33823 | 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.) |
β’ (πΉ β (dom volMblFnMπ β) β πΉ β MblFn) | ||
Theorem | mbfmcnt 33824 | All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
β’ (π β π β (π« πMblFnMπ β) = (β βm π)) | ||
Theorem | br2base 33825* | The base set for the generator of the Borel sigma-algebra on (β Γ β) is indeed (β Γ β). (Contributed by Thierry Arnoux, 22-Sep-2017.) |
β’ βͺ ran (π₯ β π β, π¦ β π β β¦ (π₯ Γ π¦)) = (β Γ β) | ||
Theorem | dya2ub 33826 | An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
β’ (π β β+ β (1 / (2β(ββ(1 β (2 logb π ))))) < π ) | ||
Theorem | sxbrsigalem0 33827* | The closed half-spaces of (β Γ β) cover (β Γ β). (Contributed by Thierry Arnoux, 11-Oct-2017.) |
β’ βͺ (ran (π β β β¦ ((π[,)+β) Γ β)) βͺ ran (π β β β¦ (β Γ (π[,)+β)))) = (β Γ β) | ||
Theorem | sxbrsigalem3 33828* | The sigma-algebra generated by the closed half-spaces of (β Γ β) is a subset of the sigma-algebra generated by the closed sets of (β Γ β). (Contributed by Thierry Arnoux, 11-Oct-2017.) |
β’ π½ = (topGenβran (,)) β β’ (sigaGenβ(ran (π β β β¦ ((π[,)+β) Γ β)) βͺ ran (π β β β¦ (β Γ (π[,)+β))))) β (sigaGenβ(Clsdβ(π½ Γt π½))) | ||
Theorem | dya2iocival 33829* | The function πΌ returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25516. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) β β’ ((π β β€ β§ π β β€) β (ππΌπ) = ((π / (2βπ))[,)((π + 1) / (2βπ)))) | ||
Theorem | dya2iocress 33830* | Dyadic intervals are subsets of β. (Contributed by Thierry Arnoux, 18-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) β β’ ((π β β€ β§ π β β€) β (ππΌπ) β β) | ||
Theorem | dya2iocbrsiga 33831* | Dyadic intervals are Borel sets of β. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) β β’ ((π β β€ β§ π β β€) β (ππΌπ) β π β) | ||
Theorem | dya2icobrsiga 33832* | Dyadic intervals are Borel sets of β. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) β β’ ran πΌ β π β | ||
Theorem | dya2icoseg 33833* | For any point and any closed-below, open-above interval of β 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (ββ(1 β (2 logb π·))) β β’ ((π β β β§ π· β β+) β βπ β ran πΌ(π β π β§ π β ((π β π·)(,)(π + π·)))) | ||
Theorem | dya2icoseg2 33834* | For any point and any open interval of β 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) β β’ ((π β β β§ πΈ β ran (,) β§ π β πΈ) β βπ β ran πΌ(π β π β§ π β πΈ)) | ||
Theorem | dya2iocrfn 33835* | The function returning dyadic square covering for a given size has domain (ran πΌ Γ ran πΌ). (Contributed by Thierry Arnoux, 19-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ π Fn (ran πΌ Γ ran πΌ) | ||
Theorem | dya2iocct 33836* | The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ ran π βΌ Ο | ||
Theorem | dya2iocnrect 33837* | For any point of an open rectangle in (β Γ β), 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) & β’ π΅ = ran (π β ran (,), π β ran (,) β¦ (π Γ π)) β β’ ((π β (β Γ β) β§ π΄ β π΅ β§ π β π΄) β βπ β ran π (π β π β§ π β π΄)) | ||
Theorem | dya2iocnei 33838* | For any point of an open set of the usual topology on (β Γ β) 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ ((π΄ β (π½ Γt π½) β§ π β π΄) β βπ β ran π (π β π β§ π β π΄)) | ||
Theorem | dya2iocuni 33839* | Every open set of (β Γ β) is a union of closed-below open-above dyadic rational rectangular subsets of (β Γ β). This union must be a countable union by dya2iocct 33836. (Contributed by Thierry Arnoux, 18-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ (π΄ β (π½ Γt π½) β βπ β π« ran π βͺ π = π΄) | ||
Theorem | dya2iocucvr 33840* | The dyadic rectangular set collection covers (β Γ β). (Contributed by Thierry Arnoux, 18-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ βͺ ran π = (β Γ β) | ||
Theorem | sxbrsigalem1 33841* | The Borel algebra on (β Γ β) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (β Γ β). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ (sigaGenβ(π½ Γt π½)) β (sigaGenβran π ) | ||
Theorem | sxbrsigalem2 33842* | The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (β Γ β) is a subset of the sigma-algebra generated by the closed half-spaces of (β Γ β). 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ (sigaGenβran π ) β (sigaGenβ(ran (π β β β¦ ((π[,)+β) Γ β)) βͺ ran (π β β β¦ (β Γ (π[,)+β))))) | ||
Theorem | sxbrsigalem4 33843* | The Borel algebra on (β Γ β) is generated by the dyadic closed-below, open-above rectangular subsets of (β Γ β). 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.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ (sigaGenβ(π½ Γt π½)) = (sigaGenβran π ) | ||
Theorem | sxbrsigalem5 33844* | First direction for sxbrsiga 33846. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
β’ π½ = (topGenβran (,)) & β’ πΌ = (π₯ β β€, π β β€ β¦ ((π₯ / (2βπ))[,)((π₯ + 1) / (2βπ)))) & β’ π = (π’ β ran πΌ, π£ β ran πΌ β¦ (π’ Γ π£)) β β’ (sigaGenβ(π½ Γt π½)) β (π β Γs π β) | ||
Theorem | sxbrsigalem6 33845 | First direction for sxbrsiga 33846, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.) |
β’ π½ = (topGenβran (,)) β β’ (sigaGenβ(π½ Γt π½)) β (π β Γs π β) | ||
Theorem | sxbrsiga 33846 | The product sigma-algebra (π β Γs π β) is the Borel algebra on (β Γ β) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.) |
β’ π½ = (topGenβran (,)) β β’ (π β Γs π β) = (sigaGenβ(π½ Γt π½)) | ||
In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure π . 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 33853) - it is monotone (omsmon 33854) - it is countably sub-additive (omssubadd 33856) See Definition 1.11.1 of [Bogachev] p. 41. | ||
Syntax | coms 33847 | Class declaration for the outer measure construction function. |
class toOMeas | ||
Definition | df-oms 33848* | Define a function constructing an outer measure. See omsval 33849 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
β’ toOMeas = (π β V β¦ (π β π« βͺ dom π β¦ inf(ran (π₯ β {π§ β π« dom π β£ (π β βͺ π§ β§ π§ βΌ Ο)} β¦ Ξ£*π¦ β π₯(πβπ¦)), (0[,]+β), < ))) | ||
Theorem | omsval 33849* | 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.) |
β’ (π β V β (toOMeasβπ ) = (π β π« βͺ dom π β¦ inf(ran (π₯ β {π§ β π« dom π β£ (π β βͺ π§ β§ π§ βΌ Ο)} β¦ Ξ£*π¦ β π₯(π βπ¦)), (0[,]+β), < ))) | ||
Theorem | omsfval 33850* | Value of the outer measure evaluated for a given set π΄. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
β’ ((π β π β§ π :πβΆ(0[,]+β) β§ π΄ β βͺ π) β ((toOMeasβπ )βπ΄) = inf(ran (π₯ β {π§ β π« dom π β£ (π΄ β βͺ π§ β§ π§ βΌ Ο)} β¦ Ξ£*π¦ β π₯(π βπ¦)), (0[,]+β), < )) | ||
Theorem | omscl 33851* | A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.) |
β’ ((π β π β§ π :πβΆ(0[,]+β) β§ π΄ β π« βͺ dom π ) β ran (π₯ β {π§ β π« dom π β£ (π΄ β βͺ π§ β§ π§ βΌ Ο)} β¦ Ξ£*π¦ β π₯(π βπ¦)) β (0[,]+β)) | ||
Theorem | omsf 33852 | A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
β’ ((π β π β§ π :πβΆ(0[,]+β)) β (toOMeasβπ ):π« βͺ dom π βΆ(0[,]+β)) | ||
Theorem | oms0 33853 | A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
β’ π = (toOMeasβπ ) & β’ (π β π β π) & β’ (π β π :πβΆ(0[,]+β)) & β’ (π β β β dom π ) & β’ (π β (π ββ ) = 0) β β’ (π β (πββ ) = 0) | ||
Theorem | omsmon 33854 | 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.) |
β’ π = (toOMeasβπ ) & β’ (π β π β π) & β’ (π β π :πβΆ(0[,]+β)) & β’ (π β π΄ β π΅) & β’ (π β π΅ β βͺ π) β β’ (π β (πβπ΄) β€ (πβπ΅)) | ||
Theorem | omssubaddlem 33855* | For any small margin πΈ, we can find a covering approaching the outer measure of a set π΄ by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
β’ π = (toOMeasβπ ) & β’ (π β π β π) & β’ (π β π :πβΆ(0[,]+β)) & β’ (π β π΄ β βͺ π) & β’ (π β (πβπ΄) β β) & β’ (π β πΈ β β+) β β’ (π β βπ₯ β {π§ β π« dom π β£ (π΄ β βͺ π§ β§ π§ βΌ Ο)}Ξ£*π€ β π₯(π βπ€) < ((πβπ΄) + πΈ)) | ||
Theorem | omssubadd 33856* | 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.) |
β’ π = (toOMeasβπ ) & β’ (π β π β π) & β’ (π β π :πβΆ(0[,]+β)) & β’ ((π β§ π¦ β π) β π΄ β βͺ π) & β’ (π β π βΌ Ο) β β’ (π β (πββͺ π¦ β π π΄) β€ Ξ£*π¦ β π(πβπ΄)) | ||
Syntax | ccarsg 33857 | Class declaration for the Caratheodory sigma-Algebra construction. |
class toCaraSiga | ||
Definition | df-carsg 33858* | Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 33859 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ toCaraSiga = (π β V β¦ {π β π« βͺ dom π β£ βπ β π« βͺ dom π((πβ(π β© π)) +π (πβ(π β π))) = (πβπ)}) | ||
Theorem | carsgval 33859* | Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) β β’ (π β (toCaraSigaβπ) = {π β π« π β£ βπ β π« π((πβ(π β© π)) +π (πβ(π β π))) = (πβπ)}) | ||
Theorem | carsgcl 33860 | Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) β β’ (π β (toCaraSigaβπ) β π« π) | ||
Theorem | elcarsg 33861* | Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) β β’ (π β (π΄ β (toCaraSigaβπ) β (π΄ β π β§ βπ β π« π((πβ(π β© π΄)) +π (πβ(π β π΄))) = (πβπ)))) | ||
Theorem | baselcarsg 33862 | The universe set, π, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) β β’ (π β π β (toCaraSigaβπ)) | ||
Theorem | 0elcarsg 33863 | The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) β β’ (π β β β (toCaraSigaβπ)) | ||
Theorem | carsguni 33864 | The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) β β’ (π β βͺ (toCaraSigaβπ) = π) | ||
Theorem | elcarsgss 33865 | Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β (toCaraSigaβπ)) β β’ (π β π΄ β π) | ||
Theorem | difelcarsg 33866 | The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β (toCaraSigaβπ)) β β’ (π β (π β π΄) β (toCaraSigaβπ)) | ||
Theorem | inelcarsg 33867* | The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ ((π β§ π β π« π β§ π β π« π) β (πβ(π βͺ π)) β€ ((πβπ) +π (πβπ))) & β’ (π β π΅ β (toCaraSigaβπ)) β β’ (π β (π΄ β© π΅) β (toCaraSigaβπ)) | ||
Theorem | unelcarsg 33868* | The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ ((π β§ π β π« π β§ π β π« π) β (πβ(π βͺ π)) β€ ((πβπ) +π (πβπ))) & β’ (π β π΅ β (toCaraSigaβπ)) β β’ (π β (π΄ βͺ π΅) β (toCaraSigaβπ)) | ||
Theorem | difelcarsg2 33869* | The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ ((π β§ π β π« π β§ π β π« π) β (πβ(π βͺ π)) β€ ((πβπ) +π (πβπ))) & β’ (π β π΅ β (toCaraSigaβπ)) β β’ (π β (π΄ β π΅) β (toCaraSigaβπ)) | ||
Theorem | carsgmon 33870* | Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β π΄ β π΅) & β’ (π β π΅ β π« π) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) β β’ (π β (πβπ΄) β€ (πβπ΅)) | ||
Theorem | carsgsigalem 33871* | Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) β β’ ((π β§ π β π« π β§ π β π« π) β (πβ(π βͺ π)) β€ ((πβπ) +π (πβπ))) | ||
Theorem | fiunelcarsg 33872* | The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ (π β π΄ β Fin) & β’ (π β π΄ β (toCaraSigaβπ)) β β’ (π β βͺ π΄ β (toCaraSigaβπ)) | ||
Theorem | carsgclctunlem1 33873* | Lemma for carsgclctun 33877. (Contributed by Thierry Arnoux, 23-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ (π β π΄ β Fin) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ (π β Disj π¦ β π΄ π¦) & β’ (π β πΈ β π« π) β β’ (π β (πβ(πΈ β© βͺ π΄)) = Ξ£*π¦ β π΄(πβ(πΈ β© π¦))) | ||
Theorem | carsggect 33874* | The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ (π β Β¬ β β π΄) & β’ (π β π΄ βΌ Ο) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ (π β Disj π¦ β π΄ π¦) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) β β’ (π β Ξ£*π§ β π΄(πβπ§) β€ (πββͺ π΄)) | ||
Theorem | carsgclctunlem2 33875* | Lemma for carsgclctun 33877. (Contributed by Thierry Arnoux, 25-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) & β’ (π β Disj π β β π΄) & β’ ((π β§ π β β) β π΄ β (toCaraSigaβπ)) & β’ (π β πΈ β π« π) & β’ (π β (πβπΈ) β +β) β β’ (π β ((πβ(πΈ β© βͺ π β β π΄)) +π (πβ(πΈ β βͺ π β β π΄))) β€ (πβπΈ)) | ||
Theorem | carsgclctunlem3 33876* | Lemma for carsgclctun 33877. (Contributed by Thierry Arnoux, 24-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) & β’ (π β π΄ βΌ Ο) & β’ (π β π΄ β (toCaraSigaβπ)) & β’ (π β πΈ β π« π) β β’ (π β ((πβ(πΈ β© βͺ π΄)) +π (πβ(πΈ β βͺ π΄))) β€ (πβπΈ)) | ||
Theorem | carsgclctun 33877* | The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) & β’ (π β π΄ βΌ Ο) & β’ (π β π΄ β (toCaraSigaβπ)) β β’ (π β βͺ π΄ β (toCaraSigaβπ)) | ||
Theorem | carsgsiga 33878* | 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.) |
β’ (π β π β π) & β’ (π β π:π« πβΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ π₯ βΌ Ο β§ π₯ β π« π) β (πββͺ π₯) β€ Ξ£*π¦ β π₯(πβπ¦)) & β’ ((π β§ π₯ β π¦ β§ π¦ β π« π) β (πβπ₯) β€ (πβπ¦)) β β’ (π β (toCaraSigaβπ) β (sigAlgebraβπ)) | ||
Theorem | omsmeas 33879 | 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.) |
β’ π = (toOMeasβπ ) & β’ π = (toCaraSigaβπ) & β’ (π β π β π) & β’ (π β π :πβΆ(0[,]+β)) & β’ (π β β β dom π ) & β’ (π β (π ββ ) = 0) β β’ (π β (π βΎ π) β (measuresβπ)) | ||
Theorem | pmeasmono 33880* | This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
β’ (π β π:π βΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ (π₯ βΌ Ο β§ π₯ β π β§ Disj π¦ β π₯ π¦)) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦)) & β’ (π β π΄ β π ) & β’ (π β π΅ β π ) & β’ (π β (π΅ β π΄) β π ) & β’ (π β π΄ β π΅) β β’ (π β (πβπ΄) β€ (πβπ΅)) | ||
Theorem | pmeasadd 33881* | 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.) |
β’ (π β π:π βΆ(0[,]+β)) & β’ (π β (πββ ) = 0) & β’ ((π β§ (π₯ βΌ Ο β§ π₯ β π β§ Disj π¦ β π₯ π¦)) β (πββͺ π₯) = Ξ£*π¦ β π₯(πβπ¦)) & β’ π = {π β π« π« π β£ (β β π β§ βπ₯ β π βπ¦ β π ((π₯ βͺ π¦) β π β§ (π₯ β π¦) β π ))} & β’ (π β π β π) & β’ (π β π΄ βΌ Ο) & β’ ((π β§ π β π΄) β π΅ β π ) & β’ (π β Disj π β π΄ π΅) β β’ (π β (πββͺ π β π΄ π΅) = Ξ£*π β π΄(πβπ΅)) | ||
Theorem | itgeq12dv 33882* | Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
β’ (π β π΄ = π΅) & β’ ((π β§ π₯ β π΄) β πΆ = π·) β β’ (π β β«π΄πΆ dπ₯ = β«π΅π· dπ₯) | ||
Syntax | citgm 33883 | Extend class notation with the (measure) Bochner integral. |
class itgm | ||
Syntax | csitm 33884 | Extend class notation with the integral metric for simple functions. |
class sitm | ||
Syntax | csitg 33885 | Extend class notation with the integral of simple functions. |
class sitg | ||
Definition | df-sitg 33886* |
Define the integral of simple functions from a measurable space
dom π to a generic space π€
equipped with the right scalar
product. π€ 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 π β Fin in the definition. Moreover, for each π₯, the pre-image (β‘π β {π₯}) is requested to be measurable, of finite measure. In this definition, (sigaGenβ(TopOpenβπ€)) is the Borel sigma-algebra on π€, and the functions π range over the measurable functions over that Borel algebra. Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.) |
β’ sitg = (π€ β V, π β βͺ ran measures β¦ (π β {π β (dom πMblFnM(sigaGenβ(TopOpenβπ€))) β£ (ran π β Fin β§ βπ₯ β (ran π β {(0gβπ€)})(πβ(β‘π β {π₯})) β (0[,)+β))} β¦ (π€ Ξ£g (π₯ β (ran π β {(0gβπ€)}) β¦ (((βHomβ(Scalarβπ€))β(πβ(β‘π β {π₯})))( Β·π βπ€)π₯))))) | ||
Definition | df-sitm 33887* | Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in β*, the range structure for this integral is (β*π βΎs (0[,]+β)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ sitm = (π€ β V, π β βͺ ran measures β¦ (π β dom (π€sitgπ), π β dom (π€sitgπ) β¦ (((β*π βΎs (0[,]+β))sitgπ)β(π βf (distβπ€)π)))) | ||
Theorem | sitgval 33888* | Value of the simple function integral builder for a given space π and measure π. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) β β’ (π β (πsitgπ) = (π β {π β (dom πMblFnMπ) β£ (ran π β Fin β§ βπ₯ β (ran π β { 0 })(πβ(β‘π β {π₯})) β (0[,)+β))} β¦ (π Ξ£g (π₯ β (ran π β { 0 }) β¦ ((π»β(πβ(β‘π β {π₯}))) Β· π₯))))) | ||
Theorem | issibf 33889* | The predicate "πΉ is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) β β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) | ||
Theorem | sibf0 33890 | The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β π β TopSp) & β’ (π β π β Mnd) β β’ (π β (βͺ dom π Γ { 0 }) β dom (πsitgπ)) | ||
Theorem | sibfmbl 33891 | A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) β β’ (π β πΉ β (dom πMblFnMπ)) | ||
Theorem | sibff 33892 | A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) β β’ (π β πΉ:βͺ dom πβΆβͺ π½) | ||
Theorem | sibfrn 33893 | A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) β β’ (π β ran πΉ β Fin) | ||
Theorem | sibfima 33894 | Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) β β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) | ||
Theorem | sibfinima 33895 | The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) & β’ (π β πΊ β dom (πsitgπ)) & β’ (π β π β TopSp) & β’ (π β π½ β Fre) β β’ (((π β§ π β ran πΉ β§ π β ran πΊ) β§ (π β 0 β¨ π β 0 )) β (πβ((β‘πΉ β {π}) β© (β‘πΊ β {π}))) β (0[,)+β)) | ||
Theorem | sibfof 33896 | 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.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) & β’ πΆ = (BaseβπΎ) & β’ (π β π β TopSp) & β’ (π β + :(π΅ Γ π΅)βΆπΆ) & β’ (π β πΊ β dom (πsitgπ)) & β’ (π β πΎ β TopSp) & β’ (π β π½ β Fre) & β’ (π β ( 0 + 0 ) = (0gβπΎ)) β β’ (π β (πΉ βf + πΊ) β dom (πΎsitgπ)) | ||
Theorem | sitgfval 33897* | Value of the Bochner integral for a simple function πΉ. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) β β’ (π β ((πsitgπ)βπΉ) = (π Ξ£g (π₯ β (ran πΉ β { 0 }) β¦ ((π»β(πβ(β‘πΉ β {π₯}))) Β· π₯)))) | ||
Theorem | sitgclg 33898* | Closure of the Bochner integral on simple functions, generic version. See sitgclbn 33899 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) & β’ πΊ = (Scalarβπ) & β’ π· = ((distβπΊ) βΎ ((BaseβπΊ) Γ (BaseβπΊ))) & β’ (π β π β TopSp) & β’ (π β π β CMnd) & β’ (π β (Scalarβπ) β βExt ) & β’ ((π β§ π β (π» β (0[,)+β)) β§ π₯ β π΅) β (π Β· π₯) β π΅) β β’ (π β ((πsitgπ)βπΉ) β π΅) | ||
Theorem | sitgclbn 33899 | 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.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) & β’ (π β π β Ban) & β’ (π β (Scalarβπ) β βExt ) β β’ (π β ((πsitgπ)βπΉ) β π΅) | ||
Theorem | sitgclcn 33900 | Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ π½ = (TopOpenβπ) & β’ π = (sigaGenβπ½) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ π» = (βHomβ(Scalarβπ)) & β’ (π β π β π) & β’ (π β π β βͺ ran measures) & β’ (π β πΉ β dom (πsitgπ)) & β’ (π β π β Ban) & β’ (π β (Scalarβπ) = βfld) β β’ (π β ((πsitgπ)βπΉ) β π΅) |
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