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affeldt-aist merged 1 commit into from
Aug 16, 2025
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simplify lemma lebesgue_measure_rat #1700

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -54,6 +54,9 @@
- in `classical_orders.v`:
+ lemma `big_lexi_order_prefix_closed_itv`

- in `lebesgue_measure.v`:
+ lemma `countable_lebesgue_measure0`

### Changed

- moved from `pi_irrational.v` to `reals.v` and changed
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28 changes: 22 additions & 6 deletions theories/lebesgue_measure.v
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Expand Up @@ -654,6 +654,26 @@ rewrite measureU //; apply/seteqP; split => // x []/=.
by rewrite in_itv/= => + xa; rewrite xa ltxx andbF.
Qed.

Lemma countable_lebesgue_measure0 (A : set R) :
countable A -> lebesgue_measure A = 0.
Proof.
move/countable_injP => [f injf].
rewrite -(injpinv_image (fun=> 0) injf).
rewrite [X in lebesgue_measure X](_ : _ = \bigcup_(x in f @` A)
[set 'pinv_(fun=> 0) A f x]); last first.
rewrite eqEsubset; split => [r [n]|r [n]].
by move=> [t At ftn] Afnr; exists n => //=; exists t.
by move=> [t At ftn] /= rAfn; exists n => //=; exists t.
rewrite measure_bigcup/=; last 2 first.
by move=> ? _; exact: measurable_set1.
move=> i j [r Ar <-] [s As <-].
by rewrite !pinvKV ?inE// => -[x [/= <- <-]].
apply: lim_near_cst => //.
apply/nearW => n.
under eq_bigr do rewrite lebesgue_measure_set1.
by rewrite big_const_idem//= addr0.
Qed.

Let lebesgue_measure_itvoo (a b : R) :
(lebesgue_measure (`]a, b[ : set R) = wlength idfun `]a, b[)%classic.
Proof.
Expand Down Expand Up @@ -801,12 +821,8 @@ rewrite (_ : range _ = \bigcup_n [set ratr (f1 n)]); last first.
apply/seteqP; split => [_ [q _ <-]|_ [n _ /= ->]]; last by exists (f1 n).
exists (f q) => //=; rewrite /f1 pinvKV// ?in_setE// => x y _ _.
by apply: bij_inj; rewrite -setTT_bijective.
rewrite measure_bigcup//; last first.
apply/trivIsetP => i j _ _ ij; apply/seteqP; split => //= _ [/= ->].
move=> /fmorph_inj.
have /set_bij_inj /[apply] := bijpinv_bij (fun=> 0) bijf.
by rewrite in_setE => /(_ Logic.I Logic.I); exact/eqP.
by rewrite eseries0// => n _ _; exact: lebesgue_measure_set1.
apply/countable_lebesgue_measure0/bigcup_countable => //.
exact: card_lexx.
Qed.

Section negligible_outer_measure.
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