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countable is measurable #1381

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Nov 6, 2024
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countable is measurable #1381

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countable is measurable
affeldt-aist Nov 5, 2024
21f03d0
gen
affeldt-aist Nov 5, 2024
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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -67,6 +67,8 @@
- in file `separation_axioms.v`,
+ new lemma `sigT_hausdorff`.

- in `measure.v`:
+ lemma `countable_measurable`

### Changed

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11 changes: 11 additions & 0 deletions theories/measure.v
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Expand Up @@ -1367,6 +1367,17 @@ Proof. by move=> PF; apply: bigcap_measurable => //; exists 1. Qed.

End sigmaring_lemmas.

Lemma countable_measurable d (T : sigmaRingType d) (A : set T) :
(forall t : T, measurable [set t]) -> countable A -> measurable A.
Proof.
move=> m1; have [->//|/set0P[r Ar]/countable_injP[f injf]] := eqVneq A set0.
rewrite -(injpinv_image (cst r) injf).
rewrite [X in _ X](_ : _ = \bigcup_(x in f @` A) [set 'pinv_(cst r) A f x]).
by apply: bigcup_measurable => _ /= [s As <-].
by rewrite eqEsubset; split=> [_ [_ [s As <-]] <-|_ [_ [s As <-]] ->];
exists (f s).
Qed.

Section sigma_ring_lambda_system.
Context d (T : sigmaRingType d).

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