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Theorem List for Metamath Proof Explorer - 40201-40300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremovolval4lem2 40201* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 40198, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovolval4 40202* The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 40198, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovolval5lem1 40203* |- ( ph -> ( sum^ (𝑛 ∈ ℕ ↦ (vol ( ( A - ( W / ( 2 ^ n ) ) ) (,) B ) ) ) ) <_ ( ( sum^ (𝑛 ∈ ℕ ↦ (vol ( A [,) B ) ) ) ) +e W ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ+)    &   𝐶 = {𝑛 ∈ ℕ ∣ 𝐴 < 𝐵}       (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +𝑒 𝑊))

Theoremovolval5lem2 40204* |- ( ( ph /\ n e. NN ) -> <. ( ( 1st (𝐹 n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd (𝐹 n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   (𝜑𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    &   𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))    &   (𝜑𝐹:ℕ⟶(ℝ × ℝ))    &   (𝜑𝐴 ran ([,) ∘ 𝐹))    &   (𝜑𝑊 ∈ ℝ+)    &   𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)       (𝜑 → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))

Theoremovolval5lem3 40205* 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* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    &   𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}       inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )

Theoremovolval5 40206* 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.)
(𝜑𝐴 ⊆ ℝ)    &   𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}       (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Theoremovnovollem1 40207* if 𝐹 is a cover of 𝐵 in , then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))    &   𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})    &   (𝜑𝐵 ran ([,) ∘ 𝐹))    &   (𝜑𝐵𝑊)    &   (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))       (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))

Theoremovnovollem2 40208* if 𝐼 is a cover of (𝐵𝑚 {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))    &   (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))    &   (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))    &   𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))       (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))

Theoremovnovollem3 40209* The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}       (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))

Theoremovnovol 40210 The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)       (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))

Theoremvonvolmbllem 40211* If a subset 𝐵 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.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦𝐵)) +𝑒 (vol*‘(𝑦𝐵))))    &   (𝜑𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵𝑚 {𝐴})))) = ((voln*‘{𝐴})‘𝑋))

Theoremvonvolmbl 40212 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.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ⊆ ℝ)       (𝜑 → ((𝐵𝑚 {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Theoremvonvol 40213 The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ dom vol)       (𝜑 → ((voln‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol‘𝐵))

Theoremvonvolmbl2 40214* A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection 𝑌 on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝑌    &   (𝜑𝐴𝑉)    &   (𝜑𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol))

Theoremvonvol2 40215* The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝑌    &   (𝜑𝐴𝑉)    &   (𝜑𝑋 ∈ dom (voln‘{𝐴}))    &   𝑌 = 𝑓𝑋 ran 𝑓       (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌))

Theoremhoimbl2 40216* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑X𝑘𝑋 (𝐴[,)𝐵) ∈ 𝑆)

Theoremvoncl 40217 The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴𝑆)       (𝜑 → ((voln‘𝑋)‘𝐴) ∈ (0[,]+∞))

Theoremvonhoi 40218* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))

Theoremvonxrcl 40219 The Lebesgue measure of a set is an extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴𝑆)       (𝜑 → ((voln‘𝑋)‘𝐴) ∈ ℝ*)

Theoremioosshoi 40220 A n-dimensional open interval is a subset of the half-open interval with the same bounds. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
X𝑘𝑋 (𝐴(,)𝐵) ⊆ X𝑘𝑋 (𝐴[,)𝐵)

Theoremvonn0hoi 40221* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremvon0val 40222 The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ dom (voln‘∅))       (𝜑 → ((voln‘∅)‘𝐴) = 0)

Theoremvonhoire 40223* The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 (𝐴[,)𝐵)) ∈ ℝ)

Theoremiinhoiicclem 40224* 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.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   (𝜑𝐹 𝑛 ∈ ℕ X𝑘𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))))       (𝜑𝐹X𝑘𝑋 (𝐴[,]𝐵))

Theoremiinhoiicc 40225* A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 𝑛 ∈ ℕ X𝑘𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))) = X𝑘𝑋 (𝐴[,]𝐵))

Theoremiunhoiioolem 40226* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)    &   (𝜑𝐹X𝑘𝑋 (𝐴(,)𝐵))    &   𝐶 = inf(ran (𝑘𝑋 ↦ ((𝐹𝑘) − 𝐴)), ℝ, < )       (𝜑𝐹 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Theoremiunhoiioo 40227* A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)       (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) = X𝑘𝑋 (𝐴(,)𝐵))

Theoremioovonmbl 40228* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ 𝑆)

Theoremiccvonmbllem 40229* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑖𝑋 ↦ ((𝐴𝑖) − (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑖𝑋 ↦ ((𝐵𝑖) + (1 / 𝑛))))       (𝜑X𝑖𝑋 ((𝐴𝑖)[,](𝐵𝑖)) ∈ 𝑆)

Theoremiccvonmbl 40230* 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.)
(𝜑𝑋 ∈ Fin)    &   𝑆 = dom (voln‘𝑋)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)[,](𝐵𝑖)) ∈ 𝑆)

Theoremvonioolem1 40231* The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))    &   𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))    &   𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑛)‘𝑘)))    &   𝐸 = inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < )    &   𝑁 = ((⌊‘(1 / 𝐸)) + 1)    &   𝑍 = (ℤ𝑁)       (𝜑𝑆 ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonioolem2 40232* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonioo 40233* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))

Theoremvonicclem1 40234* The sequence of the measures of the half-open intervals converges to the measure of their intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))    &   𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))       (𝜑𝑆 ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonicclem2 40235* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))    &   𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))

Theoremvonicc 40236* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))

Theoremsnvonmbl 40237 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑 → {𝐴} ∈ dom (voln‘𝑋))

Theoremvonn0ioo 40238* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))

Theoremvonn0icc 40239* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,](𝐵𝑘))))

Theoremctvonmbl 40240 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐴 ≼ ω)       (𝜑𝐴 ∈ dom (voln‘𝑋))

Theoremvonn0ioo2 40241* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   𝐼 = X𝑘𝑋 (𝐴(,)𝐵)       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘(𝐴(,)𝐵)))

Theoremvonsn 40242 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln‘𝑋)‘{𝐴}) = 0)

Theoremvonn0icc2 40243* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)    &   𝐼 = X𝑘𝑋 (𝐴[,]𝐵)       (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 (vol‘(𝐴[,]𝐵)))

Theoremvonct 40244 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐴 ≼ ω)       (𝜑 → ((voln‘𝑋)‘𝐴) = 0)

Theoremvitali2 40245 There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 9277). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
dom vol ⊊ 𝒫 ℝ

20.31.19.6  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 w.r.t. df-mbf 23328 are also sigma-measurable, but the definition in this section considers as measurable functions, some that are not measurable w.r.t. df-mbf 23328 (see mbfpsssmf 40328 and smfmbfcex 40305).

Syntaxcsmblfn 40246 Extend class notation with the class of real-valued measurable functions w.r.t. sigma-algebras.
class SMblFn

Definitiondf-smblfn 40247* 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 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})

Theorempimltmnf2 40248* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < -∞} = ∅)

Theorempreimagelt 40249* The preimage of a right-open, unbounded below interval, is the complement of a left-close, unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})

Theorempreimalegt 40250* The preimage of a left-open, unbounded above interval, is the complement of a right-close, unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐴 ∖ {𝑥𝐴𝐵𝐶}) = {𝑥𝐴𝐶 < 𝐵})

Theorempimconstlt0 40251* Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound smaller 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.)
𝑥𝜑    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶𝐵)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)

Theorempimconstlt1 40252* 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.)
𝑥𝜑    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)

Theorempimltpnf 40253* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)

Theorempimgtpnf2 40254* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ +∞ < (𝐹𝑥)} = ∅)

Theoremsalpreimagelt 40255* If all the preimages of left-close, 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)

Theorempimrecltpos 40256 The preimage of an unbounded below, open interval, with positive upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → {𝑥𝐴 ∣ (1 / 𝐵) < 𝐶} = ({𝑥𝐴 ∣ (1 / 𝐶) < 𝐵} ∪ {𝑥𝐴𝐵 < 0}))

Theoremsalpreimalegt 40257* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶 < 𝐵} ∈ 𝑆)

Theorempimiooltgt 40258* 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.)
𝑥𝜑    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → {𝑥𝐴𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥𝐴𝐵 < 𝑅} ∩ {𝑥𝐴𝐿 < 𝐵}))

Theorempreimaicomnf 40259* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞[,)𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) < 𝐵})

Theorempimltpnf2 40260* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)

Theorempimgtmnf2 40261* 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.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → {𝑥𝐴 ∣ -∞ < (𝐹𝑥)} = 𝐴)

Theorempimdecfgtioc 40262* Given a non-increasing 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.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑𝑆𝑌)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝑌 = (𝐼𝐴))

Theorempimincfltioc 40263* Given a non decreasing 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑𝑆𝑌)    &   𝐼 = (-∞(,]𝑆)       (𝜑𝑌 = (𝐼𝐴))

Theorempimdecfgtioo 40264* Given a non decreasing 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝐹𝑥)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴𝑅 < (𝐹𝑥)}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝑌)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝑌 = (𝐼𝐴))

Theorempimincfltioo 40265* Given a non decreasing 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝑆 = sup(𝑌, ℝ*, < )    &   (𝜑 → ¬ 𝑆𝑌)    &   𝐼 = (-∞(,)𝑆)       (𝜑𝑌 = (𝐼𝐴))

Theorempreimaioomnf 40266* Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐹 “ (-∞(,)𝐵)) = {𝑥𝐴 ∣ (𝐹𝑥) < 𝐵})

Theorempreimageiingt 40267* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶𝐵} = 𝑛 ∈ ℕ {𝑥𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵})

Theorempreimaleiinlt 40268* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵𝐶} = 𝑛 ∈ ℕ {𝑥𝐴𝐵 < (𝐶 + (1 / 𝑛))})

Theorempimgtmnf 40269* 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.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)

Theorempimrecltneg 40270 The preimage of an unbounded below, open interval, with negative upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ 0)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < 0)       (𝜑 → {𝑥𝐴 ∣ (1 / 𝐵) < 𝐶} = {𝑥𝐴𝐵 ∈ ((1 / 𝐶)(,)0)})

Theoremsalpreimagtge 40271* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎 < 𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐶𝐵} ∈ 𝑆)

Theoremsalpreimaltle 40272* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵 < 𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵𝐶} ∈ 𝑆)

Theoremissmflem 40273* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))

Theoremissmf 40274* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))

Theoremsalpreimalelt 40275* If all the preimages of right-close, 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵𝑎} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)

Theoremsalpreimagtlt 40276* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐴 = 𝑆    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝑎 < 𝐵} ∈ 𝑆)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝐶} ∈ 𝑆)

Theoremsmfpreimalt 40277* 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremsmff 40278 A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐹:𝐷⟶ℝ)

Theoremsmfdmss 40279 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.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹       (𝜑𝐷 𝑆)

Theoremissmff 40280* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))))

Theoremissmfd 40281* 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.)
𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsmfpreimaltf 40282* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfdf 40283* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) < 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsssmf 40284 The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))       (𝜑 → (𝐹𝐵) ∈ (SMblFn‘𝑆))

Theoremmbfresmf 40285 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.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑 → ran 𝐹 ⊆ ℝ)    &   𝑆 = dom vol       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremcnfsmf 40286 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.)
(𝜑𝐽 ∈ Top)    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝑆 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremincsmflem 40287* A non decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝑅 ∈ ℝ*)    &   𝑌 = {𝑥𝐴 ∣ (𝐹𝑥) < 𝑅}    &   𝐶 = sup(𝑌, ℝ*, < )    &   𝐷 = (-∞(,)𝐶)    &   𝐸 = (-∞(,]𝐶)       (𝜑 → ∃𝑏𝐵 𝑌 = (𝑏𝐴))

Theoremincsmf 40288* A real-valued, non-decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))

Theoremsmfsssmf 40289 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.)
(𝜑𝑅 ∈ SAlg)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑅𝑆)    &   (𝜑𝐹 ∈ (SMblFn‘𝑅))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmflelem 40290* 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.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹    &   (𝜑𝐷 𝑆)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremissmfle 40291* 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.)
(𝜑𝑆 ∈ SAlg)    &   𝐷 = dom 𝐹       (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝑎} ∈ (𝑆t 𝐷))))

Theoremsmfpimltmpt 40292* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝑅 ∈ ℝ)       (𝜑 → {𝑥𝐴𝐵 < 𝑅} ∈ (𝑆t 𝐴))

Theoremsmfpimltxr 40293* 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.)
𝑥𝐹    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) < 𝐴} ∈ (𝑆t 𝐷))

Theoremissmfdmpt 40294* A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑎𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑎 ∈ ℝ) → {𝑥𝐴𝐵 < 𝑎} ∈ (𝑆t 𝐴))       (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))

Theoremsmfconst 40295* 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.)
𝑥𝜑    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)    &   (𝜑𝐵 ∈ ℝ)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹 ∈ (SMblFn‘𝑆))

Theoremsssmfmpt 40296* The restriction of a sigma-measurable function is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑 → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑𝐶𝐴)       (𝜑 → (𝑥𝐶𝐵) ∈ (SMblFn‘𝑆))

Theoremcnfrrnsmf 40297 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.)
(𝜑𝑋 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝑋))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐹 ∈ ((𝐽t dom 𝐹) Cn 𝐾))    &   𝐵 = (SalGen‘𝐽)       (𝜑𝐹 ∈ (SMblFn‘𝐵))

Theoremsmfid 40298* The identity function is Borel sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → (𝑥𝐴𝑥) ∈ (SMblFn‘𝐵))

Theorembormflebmf 40299 A Borel measurable function is Lebesgue measurable. Proposition 121D (a) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))    &   𝐿 = dom (voln‘𝑋)    &   (𝜑𝐹 ∈ (SMblFn‘𝐵))       (𝜑𝐹 ∈ (SMblFn‘𝐿))

Theoremsmfpreimale 40300* Given a function measurable w.r.t. to a sigma-algebra, the preimage of an closed interval unbounded below is in the subspace sigma-algebra induced by its domain. See Proposition 121B (ii) of [Fremlin1] p. 35 (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → {𝑥𝐷 ∣ (𝐹𝑥) ≤ 𝐴} ∈ (𝑆t 𝐷))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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