Skip to content

generalization of measurable_funU #913

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
proux01 merged 2 commits into from
Apr 27, 2023
Merged

generalization of measurable_funU #913

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
d788775
measurable_fun_bigcup
affeldt-aist Apr 27, 2023
c11bba9
generalize measurable_funU
affeldt-aist Apr 27, 2023
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -8,6 +8,8 @@
+ lemma `globally0`
- in `normedtype.v`:
+ lemma `lipschitz_set0`, `lipschitz_set1`
- in `measure.v`:
+ lemma `measurable_fun_bigcup`

### Changed

Expand Down
25 changes: 16 additions & 9 deletions theories/measure.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1033,17 +1033,24 @@ Proof.
by move=> mD /= Y mY; rewrite preimage_cst; case: ifPn; rewrite ?setIT ?setI0.
Qed.

Lemma measurable_funU D E (f : T1 -> T2) :
measurable D -> measurable E ->
Lemma measurable_fun_bigcup (E : (set T1)^nat) (f : T1 -> T2) :
(forall i, measurable (E i)) ->
measurable_fun (\bigcup_i E i) f <-> (forall i, measurable_fun (E i) f).
Proof.
move=> mE; split => [|mf /= _ A mA]; last first.
by rewrite setI_bigcupl; apply: bigcup_measurable => i _; exact: mf.
move=> mf i _ A /mf => /(_ (bigcup_measurable (fun k _ => mE k))).
move=> /(measurableI (E i))-/(_ (mE i)).
by rewrite setICA setIA (@setIidr _ _ (E i))//; exact: bigcup_sup.
Qed.

Lemma measurable_funU D E (f : T1 -> T2) : measurable D -> measurable E ->
measurable_fun (D `|` E) f <-> measurable_fun D f /\ measurable_fun E f.
Proof.
move=> mD mE; split=> [mDEf|[mDf mEf] mDE A mA]; last first.
by rewrite setIUl; apply: measurableU; [exact: mDf|exact: mEf].
split.
- move=> {}mD A /mDEf => /(_ (measurableU _ _ mD mE))/(measurableI D)-/(_ mD).
by rewrite setICA setIA setUK.
- move=> {}mE A /mDEf => /(_ (measurableU _ _ mD mE))/(measurableI E)-/(_ mE).
by rewrite setICA setIA setUC setUK.
move=> mD mE; rewrite -bigcup2E; apply: (iff_trans (measurable_fun_bigcup _ _)).
by move=> [//|[//|//=]].
split=> [mf|[Df Dg] [//|[//|/= _ _ Y mY]]]; last by rewrite set0I.
by split; [exact: (mf 0%N)|exact: (mf 1%N)].
Qed.

Lemma measurable_funS E D (f : T1 -> T2) :
Expand Down