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Measurable type for normed modules (generalizes PR#2016) #2017

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6526d2b
sigma-algebra for normed modules + sfun codomain sigma ring & module …
Jul 8, 2026
5e79d43
added image_indic_lmod_sub
Jul 9, 2026
de4f13a
rebase
Nov 10, 2026
bf3fdfd
small fixes
Nov 10, 2026
d5cfc55
add normedtype_measure.v
affeldt-aist Nov 12, 2026
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1 change: 1 addition & 0 deletions _CoqProject
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Expand Up @@ -116,6 +116,7 @@ theories/measure_theory/measurable_function.v
theories/measure_theory/signed_measure.v
theories/measure_theory/measure.v

theories/real_measure/normedtype_measure.v
theories/real_measure/lebesgue_stieltjes_measure.v
theories/real_measure/lebesgue_measure.v
theories/real_measure/measurable_realfun.v
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1 change: 1 addition & 0 deletions theories/Make
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Expand Up @@ -83,6 +83,7 @@ measure_theory/measurable_function.v
measure_theory/signed_measure.v
measure_theory/measure.v

real_measure/normedtype_measure.v
real_measure/lebesgue_stieltjes_measure.v
real_measure/lebesgue_measure.v
real_measure/measurable_realfun.v
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6 changes: 3 additions & 3 deletions theories/lebesgue_integral_theory/lebesgue_integral_differentiation.v
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Expand Up @@ -58,7 +58,7 @@ Section continuous_compact_integrable.
Context {rT : realType}.
Let mu : measure _ _ := @lebesgue_measure rT.
Import MeasurableR.
Let R : measurableType _ := measurableTypeR rT.
Let R : measurableType _ := (*measurableTypeR*) rT.
Local Open Scope ereal_scope.

Lemma continuous_compact_integrable (f : R -> R^o) (A : set R^o) :
Expand All @@ -78,7 +78,7 @@ Section continuous_density_L1.
Context (rT : realType).
Let mu : measure _ _ := @lebesgue_measure rT.
Import MeasurableR.
Let R : measurableType _ := measurableTypeR rT.
Let R : measurableType _ := (*measurableTypeR*) rT.
Local Open Scope ereal_scope.

Import HBSimple.
Expand Down Expand Up @@ -180,7 +180,7 @@ Section lebesgue_differentiation_continuous.
Context {rT : realType}.
Let mu : measure _ _ := @lebesgue_measure rT.
Import MeasurableR.
Let R : measurableType _ := measurableTypeR rT.
Let R : measurableType _ := (*measurableTypeR*) rT.

Let ballE (x : R) (r : {posnum rT}) :
ball x r%:num = `](x - r%:num), (x + r%:num)[%classic :> set rT.
Expand Down
6 changes: 3 additions & 3 deletions theories/lebesgue_integral_theory/lebesgue_integral_nonneg.v
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Expand Up @@ -557,7 +557,7 @@ Local Notation mu := lebesgue_measure.
Import MeasurableR.

Lemma lebesgue_measureN (A : set R) : measurable A ->
pushforward mu (-%R : _ -> measurableTypeR R) A = mu A.
pushforward mu (-%R : _ -> (*measurableTypeR*) R) A = mu A.
Proof.
move=> mA; apply/esym/lebesgue_measure_unique => //= _ [[a b]] _ <-.
rewrite /pushforward opp_preimage_itvbndbnd/= !lebesgue_measure_itv/=.
Expand All @@ -569,7 +569,7 @@ Lemma ge0_integration_by_substitution0 (f : R -> \bar R) :
\int[mu]_(r in `[0%R, +oo[) f r = \int[mu]_(r in `]-oo, 0%R]) f (- r)%R.
Proof.
move=> mf f0; rewrite (_ : `]-oo, 0%R]%classic =
(-%R : _ -> measurableTypeR R) @^-1` `[0%R, +oo[).
(-%R : _ -> (*measurableTypeR*) R) @^-1` `[0%R, +oo[).
by rewrite opp_preimage_itvbndy oppr0.
rewrite -ge0_integral_pushforward//=; first exact: measurable_funS mf.
by apply: eq_measure_integral => //= A mA AS; exact/esym/lebesgue_measureN.
Expand Down Expand Up @@ -1068,7 +1068,7 @@ End integral_patch.
Section ge0_cvgn_integral.
Local Open Scope ereal_scope.
Import MeasurableR.
Context {R : realType} (mu : {measure set (@measurableTypeR R) -> \bar R})
Context {R : realType} (mu : {measure set ((*@measurableTypeR*) R) -> \bar R})
(f : R -> R).
Hypothesis f0 : (forall x, 0 <= f x)%R.

Expand Down
2 changes: 1 addition & 1 deletion theories/lebesgue_integral_theory/measurable_fun_approximation.v
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Expand Up @@ -703,7 +703,7 @@ Hint Extern 0 (hausdorff_space _) => (exact: Rhausdorff) : core.
Local Open Scope ereal_scope.
Context (rT : realType) (A : set rT).
Let mu : measure _ _ := @lebesgue_measure rT.
Let R : measurableType _ := measurableTypeR rT.
Let R : measurableType _ := rT(*measurableTypeR rT*).
Hypothesis mA : measurable A.
Hypothesis finA : mu A < +oo.

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8 changes: 4 additions & 4 deletions theories/probability_theory/random_variable.v
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Expand Up @@ -381,7 +381,7 @@ Context {R : realType} (f : cumulativeBounded (0:R) (1:R)).
Local Open Scope measure_display_scope.
Import MeasurableR.

Let idTR : measurableTypeR R -> R := idfun.
Let idTR : (*measurableTypeR*) R -> R := idfun.

#[local] HB.instance Definition _ :=
@isMeasurableFun.Build _ _ _ _ idTR (@measurable_id _ _ setT).
Expand Down Expand Up @@ -410,10 +410,10 @@ End cdf_of_lebesgue_stieltjes_measure.

Section lebesgue_stieltjes_measure_of_cdf.
Import MeasurableR.
Context {R : realType} (P : probability (measurableTypeR R) R).
Context {R : realType} (P : probability ((*measurableTypeR*) R) R).
Local Open Scope measure_display_scope.

Let idTR : measurableTypeR R -> R := idfun.
Let idTR : (*measurableTypeR*) R -> R := idfun.

#[local] HB.instance Definition _ :=
@isMeasurableFun.Build _ _ _ _ idTR (@measurable_id _ _ setT).
Expand Down Expand Up @@ -626,7 +626,7 @@ Let mu : {measure set _ -> \bar R} := @lebesgue_measure R.
Lemma ge0_expectation_ccdf (X : {RV P >-> R}) : (forall x, 0 <= X x)%R ->
'E_P[X] = \int[mu]_(r in `[0%R, +oo[) ccdf X r.
Proof.
pose GR := measurableTypeR R.
pose GR := (*measurableTypeR*) R.
pose distrX := distribution P X.
pose D : set R := `[0%R, +oo[%classic.
move=> X_ge0.
Expand Down
17 changes: 7 additions & 10 deletions theories/real_measure/lebesgue_measure.v
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Expand Up @@ -8,7 +8,7 @@ From mathcomp Require Import boolp classical_sets functions cardinality fsbigop
reals.
From mathcomp Require Import topology ereal numfun normedtype sequences
esum measure.
From mathcomp Require Export lebesgue_stieltjes_measure measurable_realfun.
From mathcomp Require Export normedtype_measure lebesgue_stieltjes_measure measurable_realfun.

(**md**************************************************************************)
(* # Lebesgue Measure *)
Expand Down Expand Up @@ -572,9 +572,7 @@ apply: (@measure_semi_sigma_additive _ _ _ (@lebesgue_measure R)
move=> [X mX [X' mX']] XX'Fn.
apply: measurable_image_fine.
rewrite -XX'Fn.
apply: measurableU.
apply: measurable_image_EFin.
by rewrite RGenOpenSets.measurableE/= in mX.
apply: measurableU; first exact: measurable_image_EFin.
by case: mX' => //; exact: measurableU.
- move=> i j _ _ [x [[a [Fia aoo ax] [b [Fjb boo] bx]]]].
move: tF => /(_ i j Logic.I Logic.I); apply.
Expand All @@ -591,32 +589,31 @@ apply: (@measure_semi_sigma_additive _ _ _ (@lebesgue_measure R)
+ by rewrite not_orP => -[]/(_ erefl).
+ by rewrite not_orP => -[]/(_ erefl).
- move: mUF.
rewrite {1}/measurable /emeasurable /= => -[X mX [Y []]] {Y};
have moX : measurable X by rewrite -RGenOpenSets.measurableE.
rewrite {1}/measurable /emeasurable /= => -[X mX [Y []]] {Y}.
+ rewrite setU0 => h.
rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
rewrite [X in measurable X](_ : _ = X)// predeqE => r; split => [|Xr].
move=> -[n _ [x [Fnx xoo <-{r}]]].
have : (\bigcup_n F n) x by exists n.
by rewrite -h => -[x' Xx' <-].
have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; exists r.
by exists n => //; exists r%:E => //; split => // -[].
+ move=> h.
rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
rewrite [X in measurable X](_ : _ = X)// predeqE => r; split => [|Xr].
move=> -[n _ [x [Fnx xoo <-]]].
have : (\bigcup_n F n) x by exists n.
by rewrite -h => -[[x' Xx' <-//]|xoo']; move/not_orP : xoo => -[].
have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; left; exists r.
by exists n => //; exists r%:E => //; split => // -[].
+ (* NB: almost the same as the previous one, factorize?*)
move=> h.
rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
rewrite [X in measurable X](_ : _ = X)// predeqE => r; split => [|Xr].
move=> -[n _ [x [Fnx xoo <-]]].
have : (\bigcup_n F n) x by exists n.
by rewrite -h => -[[x' Xx' <-//]|xoo']; move/not_orP : xoo => -[].
have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; left; exists r.
by exists n => //; exists r%:E => //; split => // -[].
+ move=> h.
rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
rewrite [X in measurable X](_ : _ = X)// predeqE => r; split => [|Xr].
move=> -[n _ [x [Fnx xoo <-]]].
have : (\bigcup_n F n) x by exists n.
by rewrite -h => -[[x' Xx' <-//]|].
Expand Down
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