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Theorem List for Metamath Proof Explorer - 44601-44700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovolval5lem1 44601* (πœ‘ β†’ (Ξ£^β€˜(𝑛 ∈ β„• ↦ (volβ€˜((𝐴 βˆ’ (π‘Š / (2↑𝑛) ))(,)𝐡)))) ≀ ((Ξ£^β€˜(𝑛 ∈ β„• ↦ (volβ€˜(𝐴[,)𝐡) ))) +𝑒 π‘Š)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ π‘Š ∈ ℝ+)    &   πΆ = {𝑛 ∈ β„• ∣ 𝐴 < 𝐡}    β‡’   (πœ‘ β†’ (Ξ£^β€˜(𝑛 ∈ β„• ↦ (volβ€˜((𝐴 βˆ’ (π‘Š / (2↑𝑛)))(,)𝐡)))) ≀ ((Ξ£^β€˜(𝑛 ∈ β„• ↦ (volβ€˜(𝐴[,)𝐡)))) +𝑒 π‘Š))
 
Theoremovolval5lem2 44602* ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ⟨((1st β€˜(πΉβ€˜π‘›)) βˆ’ (π‘Š / (2↑𝑛))), (2nd β€˜(πΉβ€˜π‘›))⟩ ∈ (ℝ Γ— ℝ)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑄 = {𝑧 ∈ ℝ* ∣ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)))}    &   (πœ‘ β†’ π‘Œ = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝐹)))    &   π‘ = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝐺))    &   (πœ‘ β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))    &   (πœ‘ β†’ 𝐴 βŠ† βˆͺ ran ([,) ∘ 𝐹))    &   (πœ‘ β†’ π‘Š ∈ ℝ+)    &   πΊ = (𝑛 ∈ β„• ↦ ⟨((1st β€˜(πΉβ€˜π‘›)) βˆ’ (π‘Š / (2↑𝑛))), (2nd β€˜(πΉβ€˜π‘›))⟩)    β‡’   (πœ‘ β†’ βˆƒπ‘§ ∈ 𝑄 𝑧 ≀ (π‘Œ +𝑒 π‘Š))
 
Theoremovolval5lem3 44603* The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐴 βŠ† βˆͺ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝑓)))}    &   π‘„ = {𝑧 ∈ ℝ* ∣ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)))}    β‡’   inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
 
Theoremovolval5 44604* The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘€ = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐴 βŠ† βˆͺ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝑓)))}    β‡’   (πœ‘ β†’ (vol*β€˜π΄) = inf(𝑀, ℝ*, < ))
 
Theoremovnovollem1 44605* if 𝐹 is a cover of 𝐡 in ℝ, then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•))    &   πΌ = (𝑗 ∈ β„• ↦ {⟨𝐴, (πΉβ€˜π‘—)⟩})    &   (πœ‘ β†’ 𝐡 βŠ† βˆͺ ran ([,) ∘ 𝐹))    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝑍 = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝐹)))    β‡’   (πœ‘ β†’ βˆƒπ‘– ∈ (((ℝ Γ— ℝ) ↑m {𝐴}) ↑m β„•)((𝐡 ↑m {𝐴}) βŠ† βˆͺ 𝑗 ∈ β„• Xπ‘˜ ∈ {𝐴} (([,) ∘ (π‘–β€˜π‘—))β€˜π‘˜) ∧ 𝑍 = (Ξ£^β€˜(𝑗 ∈ β„• ↦ βˆπ‘˜ ∈ {𝐴} (volβ€˜(([,) ∘ (π‘–β€˜π‘—))β€˜π‘˜))))))
 
Theoremovnovollem2 44606* if 𝐼 is a cover of (𝐡 ↑m {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ (((ℝ Γ— ℝ) ↑m {𝐴}) ↑m β„•))    &   (πœ‘ β†’ (𝐡 ↑m {𝐴}) βŠ† βˆͺ 𝑗 ∈ β„• Xπ‘˜ ∈ {𝐴} (([,) ∘ (πΌβ€˜π‘—))β€˜π‘˜))    &   (πœ‘ β†’ 𝑍 = (Ξ£^β€˜(𝑗 ∈ β„• ↦ βˆπ‘˜ ∈ {𝐴} (volβ€˜(([,) ∘ (πΌβ€˜π‘—))β€˜π‘˜)))))    &   πΉ = (𝑗 ∈ β„• ↦ ((πΌβ€˜π‘—)β€˜π΄))    β‡’   (πœ‘ β†’ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐡 βŠ† βˆͺ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝑓))))
 
Theoremovnovollem3 44607* The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ)    &   π‘€ = {𝑧 ∈ ℝ* ∣ βˆƒπ‘– ∈ (((ℝ Γ— ℝ) ↑m {𝐴}) ↑m β„•)((𝐡 ↑m {𝐴}) βŠ† βˆͺ 𝑗 ∈ β„• Xπ‘˜ ∈ {𝐴} (([,) ∘ (π‘–β€˜π‘—))β€˜π‘˜) ∧ 𝑧 = (Ξ£^β€˜(𝑗 ∈ β„• ↦ βˆπ‘˜ ∈ {𝐴} (volβ€˜(([,) ∘ (π‘–β€˜π‘—))β€˜π‘˜)))))}    &   π‘ = {𝑧 ∈ ℝ* ∣ βˆƒπ‘“ ∈ ((ℝ Γ— ℝ) ↑m β„•)(𝐡 βŠ† βˆͺ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Ξ£^β€˜((vol ∘ [,)) ∘ 𝑓)))}    β‡’   (πœ‘ β†’ ((voln*β€˜{𝐴})β€˜(𝐡 ↑m {𝐴})) = (vol*β€˜π΅))
 
Theoremovnovol 44608 The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ)    β‡’   (πœ‘ β†’ ((voln*β€˜{𝐴})β€˜(𝐡 ↑m {𝐴})) = (vol*β€˜π΅))
 
Theoremvonvolmbllem 44609* If a subset 𝐡 of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with 𝑛 equal to 1). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 ℝ(vol*β€˜π‘¦) = ((vol*β€˜(𝑦 ∩ 𝐡)) +𝑒 (vol*β€˜(𝑦 βˆ– 𝐡))))    &   (πœ‘ β†’ 𝑋 βŠ† (ℝ ↑m {𝐴}))    &   π‘Œ = βˆͺ 𝑓 ∈ 𝑋 ran 𝑓    β‡’   (πœ‘ β†’ (((voln*β€˜{𝐴})β€˜(𝑋 ∩ (𝐡 ↑m {𝐴}))) +𝑒 ((voln*β€˜{𝐴})β€˜(𝑋 βˆ– (𝐡 ↑m {𝐴})))) = ((voln*β€˜{𝐴})β€˜π‘‹))
 
Theoremvonvolmbl 44610 A subset of Real numbers is Lebesgue measurable if and only if its corresponding 1-dimensional set is measurable w.r.t. the 1-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ)    β‡’   (πœ‘ β†’ ((𝐡 ↑m {𝐴}) ∈ dom (volnβ€˜{𝐴}) ↔ 𝐡 ∈ dom vol))
 
Theoremvonvol 44611 The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ dom vol)    β‡’   (πœ‘ β†’ ((volnβ€˜{𝐴})β€˜(𝐡 ↑m {𝐴})) = (volβ€˜π΅))
 
Theoremvonvolmbl2 44612* A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection π‘Œ on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘“π‘Œ    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 βŠ† (ℝ ↑m {𝐴}))    &   π‘Œ = βˆͺ 𝑓 ∈ 𝑋 ran 𝑓    β‡’   (πœ‘ β†’ (𝑋 ∈ dom (volnβ€˜{𝐴}) ↔ π‘Œ ∈ dom vol))
 
Theoremvonvol2 44613* The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘“π‘Œ    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ dom (volnβ€˜{𝐴}))    &   π‘Œ = βˆͺ 𝑓 ∈ 𝑋 ran 𝑓    β‡’   (πœ‘ β†’ ((volnβ€˜{𝐴})β€˜π‘‹) = (volβ€˜π‘Œ))
 
Theoremhoimbl2 44614* Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ Xπ‘˜ ∈ 𝑋 (𝐴[,)𝐡) ∈ 𝑆)
 
Theoremvoncl 44615 The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜π΄) ∈ (0[,]+∞))
 
Theoremvonhoi 44616* The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). A direct consequence of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,)(π΅β€˜π‘˜))    &   πΏ = (π‘₯ ∈ Fin ↦ (π‘Ž ∈ (ℝ ↑m π‘₯), 𝑏 ∈ (ℝ ↑m π‘₯) ↦ if(π‘₯ = βˆ…, 0, βˆπ‘˜ ∈ π‘₯ (volβ€˜((π‘Žβ€˜π‘˜)[,)(π‘β€˜π‘˜))))))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = (𝐴(πΏβ€˜π‘‹)𝐡))
 
Theoremvonxrcl 44617 The Lebesgue measure of a set is an extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜π΄) ∈ ℝ*)
 
Theoremioosshoi 44618 A n-dimensional open interval is a subset of the half-open interval with the same bounds. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Xπ‘˜ ∈ 𝑋 (𝐴(,)𝐡) βŠ† Xπ‘˜ ∈ 𝑋 (𝐴[,)𝐡)
 
Theoremvonn0hoi 44619* The Lebesgue outer measure of a multidimensional half-open interval when the dimension of the space is nonzero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,)(π΅β€˜π‘˜))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 (volβ€˜((π΄β€˜π‘˜)[,)(π΅β€˜π‘˜))))
 
Theoremvon0val 44620 The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ dom (volnβ€˜βˆ…))    β‡’   (πœ‘ β†’ ((volnβ€˜βˆ…)β€˜π΄) = 0)
 
Theoremvonhoire 44621* The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜Xπ‘˜ ∈ 𝑋 (𝐴[,)𝐡)) ∈ ℝ)
 
Theoremiinhoiicclem 44622* A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ∩ 𝑛 ∈ β„• Xπ‘˜ ∈ 𝑋 (𝐴[,)(𝐡 + (1 / 𝑛))))    β‡’   (πœ‘ β†’ 𝐹 ∈ Xπ‘˜ ∈ 𝑋 (𝐴[,]𝐡))
 
Theoremiinhoiicc 44623* A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ∩ 𝑛 ∈ β„• Xπ‘˜ ∈ 𝑋 (𝐴[,)(𝐡 + (1 / 𝑛))) = Xπ‘˜ ∈ 𝑋 (𝐴[,]𝐡))
 
Theoremiunhoiioolem 44624* A n-dimensional open interval expressed as the indexed union of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐹 ∈ Xπ‘˜ ∈ 𝑋 (𝐴(,)𝐡))    &   πΆ = inf(ran (π‘˜ ∈ 𝑋 ↦ ((πΉβ€˜π‘˜) βˆ’ 𝐴)), ℝ, < )    β‡’   (πœ‘ β†’ 𝐹 ∈ βˆͺ 𝑛 ∈ β„• Xπ‘˜ ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐡))
 
Theoremiunhoiioo 44625* A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ βˆͺ 𝑛 ∈ β„• Xπ‘˜ ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐡) = Xπ‘˜ ∈ 𝑋 (𝐴(,)𝐡))
 
Theoremioovonmbl 44626* Any n-dimensional open interval is Lebesgue measurable. This is the first statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„*)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„*)    β‡’   (πœ‘ β†’ X𝑖 ∈ 𝑋 ((π΄β€˜π‘–)(,)(π΅β€˜π‘–)) ∈ 𝑆)
 
Theoremiccvonmbllem 44627* Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΆ = (𝑛 ∈ β„• ↦ (𝑖 ∈ 𝑋 ↦ ((π΄β€˜π‘–) βˆ’ (1 / 𝑛))))    &   π· = (𝑛 ∈ β„• ↦ (𝑖 ∈ 𝑋 ↦ ((π΅β€˜π‘–) + (1 / 𝑛))))    β‡’   (πœ‘ β†’ X𝑖 ∈ 𝑋 ((π΄β€˜π‘–)[,](π΅β€˜π‘–)) ∈ 𝑆)
 
Theoremiccvonmbl 44628* Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π‘† = dom (volnβ€˜π‘‹)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    β‡’   (πœ‘ β†’ X𝑖 ∈ 𝑋 ((π΄β€˜π‘–)[,](π΅β€˜π‘–)) ∈ 𝑆)
 
Theoremvonioolem1 44629* The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ (π΄β€˜π‘˜) < (π΅β€˜π‘˜))    &   πΆ = (𝑛 ∈ β„• ↦ (π‘˜ ∈ 𝑋 ↦ ((π΄β€˜π‘˜) + (1 / 𝑛))))    &   π· = (𝑛 ∈ β„• ↦ Xπ‘˜ ∈ 𝑋 (((πΆβ€˜π‘›)β€˜π‘˜)[,)(π΅β€˜π‘˜)))    &   π‘† = (𝑛 ∈ β„• ↦ ((volnβ€˜π‘‹)β€˜(π·β€˜π‘›)))    &   π‘‡ = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝑋 ((π΅β€˜π‘˜) βˆ’ ((πΆβ€˜π‘›)β€˜π‘˜)))    &   πΈ = inf(ran (π‘˜ ∈ 𝑋 ↦ ((π΅β€˜π‘˜) βˆ’ (π΄β€˜π‘˜))), ℝ, < )    &   π‘ = ((βŒŠβ€˜(1 / 𝐸)) + 1)    &   π‘ = (β„€β‰₯β€˜π‘)    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 ((π΅β€˜π‘˜) βˆ’ (π΄β€˜π‘˜)))
 
Theoremvonioolem2 44630* The n-dimensional Lebesgue measure of open intervals. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ (π΄β€˜π‘˜) < (π΅β€˜π‘˜))    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)(,)(π΅β€˜π‘˜))    &   πΆ = (𝑛 ∈ β„• ↦ (π‘˜ ∈ 𝑋 ↦ ((π΄β€˜π‘˜) + (1 / 𝑛))))    &   π· = (𝑛 ∈ β„• ↦ Xπ‘˜ ∈ 𝑋 (((πΆβ€˜π‘›)β€˜π‘˜)[,)(π΅β€˜π‘˜)))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 ((π΅β€˜π‘˜) βˆ’ (π΄β€˜π‘˜)))
 
Theoremvonioo 44631* The n-dimensional Lebesgue measure of an open interval. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)(,)(π΅β€˜π‘˜))    &   πΏ = (π‘₯ ∈ Fin ↦ (π‘Ž ∈ (ℝ ↑m π‘₯), 𝑏 ∈ (ℝ ↑m π‘₯) ↦ if(π‘₯ = βˆ…, 0, βˆπ‘˜ ∈ π‘₯ (volβ€˜((π‘Žβ€˜π‘˜)[,)(π‘β€˜π‘˜))))))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = (𝐴(πΏβ€˜π‘‹)𝐡))
 
Theoremvonicclem1 44632* The sequence of the measures of the half-open intervals converges to the measure of their intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ (π΄β€˜π‘˜) ≀ (π΅β€˜π‘˜))    &   πΆ = (𝑛 ∈ β„• ↦ (π‘˜ ∈ 𝑋 ↦ ((π΅β€˜π‘˜) + (1 / 𝑛))))    &   π· = (𝑛 ∈ β„• ↦ Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,)((πΆβ€˜π‘›)β€˜π‘˜)))    &   π‘† = (𝑛 ∈ β„• ↦ ((volnβ€˜π‘‹)β€˜(π·β€˜π‘›)))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 ((π΅β€˜π‘˜) βˆ’ (π΄β€˜π‘˜)))
 
Theoremvonicclem2 44633* The n-dimensional Lebesgue measure of closed intervals. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ (π΄β€˜π‘˜) ≀ (π΅β€˜π‘˜))    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,](π΅β€˜π‘˜))    &   πΆ = (𝑛 ∈ β„• ↦ (π‘˜ ∈ 𝑋 ↦ ((π΅β€˜π‘˜) + (1 / 𝑛))))    &   π· = (𝑛 ∈ β„• ↦ Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,)((πΆβ€˜π‘›)β€˜π‘˜)))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 ((π΅β€˜π‘˜) βˆ’ (π΄β€˜π‘˜)))
 
Theoremvonicc 44634* The n-dimensional Lebesgue measure of a closed interval. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,](π΅β€˜π‘˜))    &   πΏ = (π‘₯ ∈ Fin ↦ (π‘Ž ∈ (ℝ ↑m π‘₯), 𝑏 ∈ (ℝ ↑m π‘₯) ↦ if(π‘₯ = βˆ…, 0, βˆπ‘˜ ∈ π‘₯ (volβ€˜((π‘Žβ€˜π‘˜)[,)(π‘β€˜π‘˜))))))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = (𝐴(πΏβ€˜π‘‹)𝐡))
 
Theoremsnvonmbl 44635 A n-dimensional singleton is Lebesgue measurable. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴 ∈ (ℝ ↑m 𝑋))    β‡’   (πœ‘ β†’ {𝐴} ∈ dom (volnβ€˜π‘‹))
 
Theoremvonn0ioo 44636* The n-dimensional Lebesgue measure of an open interval when the dimension of the space is nonzero. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)(,)(π΅β€˜π‘˜))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 (volβ€˜((π΄β€˜π‘˜)[,)(π΅β€˜π‘˜))))
 
Theoremvonn0icc 44637* The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   (πœ‘ β†’ 𝐴:π‘‹βŸΆβ„)    &   (πœ‘ β†’ 𝐡:π‘‹βŸΆβ„)    &   πΌ = Xπ‘˜ ∈ 𝑋 ((π΄β€˜π‘˜)[,](π΅β€˜π‘˜))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 (volβ€˜((π΄β€˜π‘˜)[,](π΅β€˜π‘˜))))
 
Theoremctvonmbl 44638 Any n-dimensional countable set is Lebesgue measurable. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴 βŠ† (ℝ ↑m 𝑋))    &   (πœ‘ β†’ 𝐴 β‰Ό Ο‰)    β‡’   (πœ‘ β†’ 𝐴 ∈ dom (volnβ€˜π‘‹))
 
Theoremvonn0ioo2 44639* The n-dimensional Lebesgue measure of an open interval when the dimension of the space is nonzero. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    &   πΌ = Xπ‘˜ ∈ 𝑋 (𝐴(,)𝐡)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 (volβ€˜(𝐴(,)𝐡)))
 
Theoremvonsn 44640 The n-dimensional Lebesgue measure of a singleton is zero. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴 ∈ (ℝ ↑m 𝑋))    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜{𝐴}) = 0)
 
Theoremvonn0icc2 44641* The n-dimensional Lebesgue measure of a closed interval, when the dimension of the space is nonzero. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐡 ∈ ℝ)    &   πΌ = Xπ‘˜ ∈ 𝑋 (𝐴[,]𝐡)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜πΌ) = βˆπ‘˜ ∈ 𝑋 (volβ€˜(𝐴[,]𝐡)))
 
Theoremvonct 44642 The n-dimensional Lebesgue measure of any countable set is zero. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   (πœ‘ β†’ 𝐴 βŠ† (ℝ ↑m 𝑋))    &   (πœ‘ β†’ 𝐴 β‰Ό Ο‰)    β‡’   (πœ‘ β†’ ((volnβ€˜π‘‹)β€˜π΄) = 0)
 
Theoremvitali2 44643 There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 10365). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
dom vol ⊊ 𝒫 ℝ
 
21.38.19.6  Measurable functions

Proofs for most of the theorems in section 121 of [Fremlin1]. Real-valued functions are considered, and measurability is defined with respect to an arbitrary sigma-algebra. When the sigma-algebra on the domain is the Lebesgue measure on the reals, then all real-valued measurable functions in the sense of df-mbf 24905 are also sigma-measurable, but the definition in this section considers as measurable functions, some that are not measurable in the sense of df-mbf 24905 (see mbfpsssmf 44732 and smfmbfcex 44709).

 
Syntaxcsmblfn 44644 Extend class notation with the class of real-valued measurable functions w.r.t. sigma-algebras.
class SMblFn
 
Definitiondf-smblfn 44645* Define a real-valued measurable function w.r.t. a given sigma-algebra. See Definition 121C of [Fremlin1] p. 36 and Definition 135E (b) of [Fremlin1] p. 80 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm βˆͺ 𝑠) ∣ βˆ€π‘Ž ∈ ℝ (◑𝑓 β€œ (-∞(,)π‘Ž)) ∈ (𝑠 β†Ύt dom 𝑓)})
 
Theorempimltmnf2f 44646 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < -∞} = βˆ…)
 
Theorempimltmnf2 44647* Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < -∞} = βˆ…)
 
Theorempreimagelt 44648* The preimage of a right-open, unbounded below interval, is the complement of a left-closed unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 βˆ– {π‘₯ ∈ 𝐴 ∣ 𝐢 ≀ 𝐡}) = {π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝐢})
 
Theorempreimalegt 44649* The preimage of a left-open, unbounded above interval, is the complement of a right-closed unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 βˆ– {π‘₯ ∈ 𝐴 ∣ 𝐡 ≀ 𝐢}) = {π‘₯ ∈ 𝐴 ∣ 𝐢 < 𝐡})
 
Theorempimconstlt0 44650* Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound less than or equal to the constant, is the empty set. Second part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΉ = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ≀ 𝐡)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝐢} = βˆ…)
 
Theorempimconstlt1 44651* Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΉ = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝐢} = 𝐴)
 
Theorempimltpnff 44652 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
β„²π‘₯πœ‘    &   β„²π‘₯𝐴    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < +∞} = 𝐴)
 
Theorempimltpnf 44653* Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < +∞} = 𝐴)
 
Theorempimgtpnf2f 44654 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2021.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘₯)} = βˆ…)
 
Theorempimgtpnf2 44655* Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ +∞ < (πΉβ€˜π‘₯)} = βˆ…)
 
Theoremsalpreimagelt 44656* If all the preimages of left-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π΄ = βˆͺ 𝑆    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ π‘Ž ≀ 𝐡} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝐢} ∈ 𝑆)
 
Theorempimrecltpos 44657 The preimage of an unbounded below, open interval, with positive upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 β‰  0)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (1 / 𝐡) < 𝐢} = ({π‘₯ ∈ 𝐴 ∣ (1 / 𝐢) < 𝐡} βˆͺ {π‘₯ ∈ 𝐴 ∣ 𝐡 < 0}))
 
Theoremsalpreimalegt 44658* If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π΄ = βˆͺ 𝑆    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 ≀ π‘Ž} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐢 < 𝐡} ∈ 𝑆)
 
Theorempimiooltgt 44659* The preimage of an open interval is the intersection of the preimage of an unbounded below open interval and an unbounded above open interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐿 ∈ ℝ*)    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 ∈ (𝐿(,)𝑅)} = ({π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝑅} ∩ {π‘₯ ∈ 𝐴 ∣ 𝐿 < 𝐡}))
 
Theorempreimaicomnf 44660* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (◑𝐹 β€œ (-∞[,)𝐡)) = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝐡})
 
Theorempimltpnf2f 44661 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
 
Theorempimltpnf2 44662* Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < +∞} = 𝐴)
 
Theorempimgtmnf2 44663* Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < (πΉβ€˜π‘₯)} = 𝐴)
 
Theorempimdecfgtioc 44664* Given a nonincreasing function, the preimage of an unbounded above, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯)))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   π‘Œ = {π‘₯ ∈ 𝐴 ∣ 𝑅 < (πΉβ€˜π‘₯)}    &   π‘† = sup(π‘Œ, ℝ*, < )    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   πΌ = (-∞(,]𝑆)    β‡’   (πœ‘ β†’ π‘Œ = (𝐼 ∩ 𝐴))
 
Theorempimincfltioc 44665* Given a nondecreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   π‘Œ = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝑅}    &   π‘† = sup(π‘Œ, ℝ*, < )    &   (πœ‘ β†’ 𝑆 ∈ π‘Œ)    &   πΌ = (-∞(,]𝑆)    β‡’   (πœ‘ β†’ π‘Œ = (𝐼 ∩ 𝐴))
 
Theorempimdecfgtioo 44666* Given a nondecreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯)))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   π‘Œ = {π‘₯ ∈ 𝐴 ∣ 𝑅 < (πΉβ€˜π‘₯)}    &   π‘† = sup(π‘Œ, ℝ*, < )    &   (πœ‘ β†’ Β¬ 𝑆 ∈ π‘Œ)    &   πΌ = (-∞(,)𝑆)    β‡’   (πœ‘ β†’ π‘Œ = (𝐼 ∩ 𝐴))
 
Theorempimincfltioo 44667* Given a nondecreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   π‘Œ = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝑅}    &   π‘† = sup(π‘Œ, ℝ*, < )    &   (πœ‘ β†’ Β¬ 𝑆 ∈ π‘Œ)    &   πΌ = (-∞(,)𝑆)    β‡’   (πœ‘ β†’ π‘Œ = (𝐼 ∩ 𝐴))
 
Theorempreimaioomnf 44668* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (◑𝐹 β€œ (-∞(,)𝐡)) = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝐡})
 
Theorempreimageiingt 44669* A preimage of a left-closed, unbounded above interval, expressed as an indexed intersection of preimages of open, unbounded above intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐢 ≀ 𝐡} = ∩ 𝑛 ∈ β„• {π‘₯ ∈ 𝐴 ∣ (𝐢 βˆ’ (1 / 𝑛)) < 𝐡})
 
Theorempreimaleiinlt 44670* A preimage of a left-open, right-closed, unbounded below interval, expressed as an indexed intersection of preimages of open, unbound below intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 ≀ 𝐢} = ∩ 𝑛 ∈ β„• {π‘₯ ∈ 𝐴 ∣ 𝐡 < (𝐢 + (1 / 𝑛))})
 
Theorempimgtmnff 44671 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
β„²π‘₯πœ‘    &   β„²π‘₯𝐴    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < 𝐡} = 𝐴)
 
Theorempimgtmnf 44672* Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ -∞ < 𝐡} = 𝐴)
 
Theorempimrecltneg 44673 The preimage of an unbounded below, open interval, with negative upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 β‰  0)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 0)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ (1 / 𝐡) < 𝐢} = {π‘₯ ∈ 𝐴 ∣ 𝐡 ∈ ((1 / 𝐢)(,)0)})
 
Theoremsalpreimagtge 44674* If all the preimages of left-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of left-closed, unbounded above intervals, belong to the sigma-algebra. (iii) implies (iv) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ π‘Ž < 𝐡} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐢 ≀ 𝐡} ∈ 𝑆)
 
Theoremsalpreimaltle 44675* If all the preimages of right-open, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-closed, unbounded below intervals, belong to the sigma-algebra. (i) implies (ii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < π‘Ž} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 ≀ 𝐢} ∈ 𝑆)
 
Theoremissmflem 44676* The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
 
Theoremissmf 44677* The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
 
Theoremsalpreimalelt 44678* If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (ii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π΄ = βˆͺ 𝑆    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 ≀ π‘Ž} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝐢} ∈ 𝑆)
 
Theoremsalpreimagtlt 44679* If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π΄ = βˆͺ 𝑆    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ π‘Ž < 𝐡} ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝐢} ∈ 𝑆)
 
Theoremsmfpreimalt 44680* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    &   π· = dom 𝐹    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
 
Theoremsmff 44681 A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ 𝐹:π·βŸΆβ„)
 
Theoremsmfdmss 44682 The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ 𝐷 βŠ† βˆͺ 𝑆)
 
Theoremissmff 44683* The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
 
Theoremissmfd 44684* A sufficient condition for "𝐹 being a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐷 βŠ† βˆͺ 𝑆)    &   (πœ‘ β†’ 𝐹:π·βŸΆβ„)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremsmfpreimaltf 44685* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    &   π· = dom 𝐹    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
 
Theoremissmfdf 44686* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯𝐹    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐷 βŠ† βˆͺ 𝑆)    &   (πœ‘ β†’ 𝐹:π·βŸΆβ„)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremsssmf 44687 The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ (SMblFnβ€˜π‘†))
 
Theoremmbfresmf 44688 A real-valued measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐹 ∈ MblFn)    &   (πœ‘ β†’ ran 𝐹 βŠ† ℝ)    &   π‘† = dom vol    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremcnfsmf 44689 A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐽 ∈ Top)    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐹 ∈ ((𝐽 β†Ύt dom 𝐹) Cn 𝐾))    &   π‘† = (SalGenβ€˜π½)    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremincsmflem 44690* A nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))    &   π½ = (topGenβ€˜ran (,))    &   π΅ = (SalGenβ€˜π½)    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   π‘Œ = {π‘₯ ∈ 𝐴 ∣ (πΉβ€˜π‘₯) < 𝑅}    &   πΆ = sup(π‘Œ, ℝ*, < )    &   π· = (-∞(,)𝐢)    &   πΈ = (-∞(,]𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 π‘Œ = (𝑏 ∩ 𝐴))
 
Theoremincsmf 44691* A real-valued, nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))    &   π½ = (topGenβ€˜ran (,))    &   π΅ = (SalGenβ€˜π½)    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π΅))
 
Theoremsmfsssmf 44692 If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑅 ∈ SAlg)    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝑅 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘…))    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremissmflelem 44693* The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   π· = dom 𝐹    &   (πœ‘ β†’ 𝐷 βŠ† βˆͺ 𝑆)    &   (πœ‘ β†’ 𝐹:π·βŸΆβ„)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) ≀ π‘Ž} ∈ (𝑆 β†Ύt 𝐷))    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremissmfle 44694* The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right-closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   π· = dom 𝐹    β‡’   (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) ≀ π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
 
Theoremsmfpimltmpt 44695* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (SMblFnβ€˜π‘†))    &   (πœ‘ β†’ 𝑅 ∈ ℝ)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < 𝑅} ∈ (𝑆 β†Ύt 𝐴))
 
Theoremsmfpimltxr 44696* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
β„²π‘₯𝐹    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))    &   π· = dom 𝐹    &   (πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
 
Theoremissmfdmpt 44697* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   β„²π‘Žπœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 < π‘Ž} ∈ (𝑆 β†Ύt 𝐴))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (SMblFnβ€˜π‘†))
 
Theoremsmfconst 44698* Given a sigma-algebra over a base set X, every partial real-valued constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝑆)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΉ = (π‘₯ ∈ 𝐴 ↦ 𝐡)    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
 
Theoremsssmfmpt 44699* The restriction of a sigma-measurable function is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑆 ∈ SAlg)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (SMblFnβ€˜π‘†))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↦ 𝐡) ∈ (SMblFnβ€˜π‘†))
 
Theoremcnfrrnsmf 44700 A function, continuous from the standard topology on the space of n-dimensional reals to the standard topology on the reals, is Borel measurable. Proposition 121D (b) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝑋 ∈ Fin)    &   π½ = (TopOpenβ€˜(ℝ^β€˜π‘‹))    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐹 ∈ ((𝐽 β†Ύt dom 𝐹) Cn 𝐾))    &   π΅ = (SalGenβ€˜π½)    β‡’   (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π΅))
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