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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremovoliunlem2 19401* Lemma for ovoliun 19403. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliunlem3 19402* Lemma for ovoliun 19403. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliun 19403* The Lebesgue outer measure function is countably sub-additive. (Many books allow as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 19383, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliun2 19404* The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 19403.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliunnul 19405* A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremshft2rab 19406* If is a shift of by , then is a shift of by . (Contributed by Mario Carneiro, 22-Mar-2014.) (Revised by Mario Carneiro, 6-Apr-2015.)

Theoremovolshftlem1 19407* Lemma for ovolshft 19409. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolshftlem2 19408* Lemma for ovolshft 19409. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolshft 19409* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremsca2rab 19410* If is a scale of by , then is a scale of by . (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolscalem1 19411* Lemma for ovolsca 19413. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremovolscalem2 19412* Lemma for ovolshft 19409. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolsca 19413* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremovolicc1 19414* The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Theoremovolicc2lem1 19415* Lemma for ovolicc2 19420. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem2 19416* Lemma for ovolicc2 19420. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem3 19417* Lemma for ovolicc2 19420. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem4 19418* Lemma for ovolicc2 19420. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem5 19419* Lemma for ovolicc2 19420. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2 19420* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc 19421 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicopnf 19422 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolre 19423 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremismbl 19424* The predicate " is Lebesgue-measurable". A set is measurable if it splits every other set in a "nice" way, that is, if the measure of the pieces and sum up to the measure of (assuming that the measure of is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremismbl2 19425* From ovolun 19397, it suffices to show that the measure of is at least the sum of the measures of and . (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremvolres 19426 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremvolf 19427 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremmblvol 19428 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblss 19429 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblsplit 19430 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremcmmbl 19431 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl 19432 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl2 19433* A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has while "outer measure zero" means that for any there is a containing with volume less than . Assuming AC, these notions are equivalent (because the intersection of all such is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremunmbl 19434 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremshftmbl 19435* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theorem0mbl 19436 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremrembl 19437 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoreminmbl 19438 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremdifmbl 19439 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremfiniunmbl 19440* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)

Theoremvolun 19441 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremvolinun 19442 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremvolfiniun 19443* The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj

Theoremiundisj 19444* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
..^

Theoremiundisj2 19445* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj ..^

Theoremvoliunlem1 19446* Lemma for voliun 19450. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem2 19447* Lemma for voliun 19450. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem3 19448* Lemma for voliun 19450. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremiunmbl 19449 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremvoliun 19450 The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Disj

Theoremvolsuplem 19451* Lemma for volsup 19452. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremvolsup 19452* The volume of the limit of an increasing sequence of measurable sets is the limit of the volumes. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremiunmbl2 19453* The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremioombl1lem1 19454* Lemma for ioombl1 19458. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem2 19455* Lemma for ioombl1 19458. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem3 19456* Lemma for ioombl1 19458. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem4 19457* Lemma for ioombl1 19458. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremioombl1 19458 An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Theoremicombl1 19459 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremicombl 19460 A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremioombl 19461 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremiccmbl 19462 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremiccvolcl 19463 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremovolioo 19464 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremovolfs2 19465 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorcl2 19466 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorf 19467 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorval 19468* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorinv2 19469* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorinv 19470* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorcl 19471* The function does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremuniiccdif 19472 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremuniioovol 19473* A disjoint union of open intervals has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 19450.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccvol 19474* An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 19450.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem1 19475* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem2a 19476* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 7-May-2015.)
Disj

Theoremuniioombllem2 19477* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj

Theoremuniioombllem3a 19478* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 8-May-2015.)
Disj

Theoremuniioombllem3 19479* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem4 19480* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem5 19481* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 25-Aug-2014.)
Disj

Theoremuniioombllem6 19482* Lemma for uniioombl 19483. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombl 19483* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 19449.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccmbl 19484* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 19449.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremdyadf 19485* The function returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadval 19486* Value of the dyadic rational function . (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadovol 19487* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadss 19488* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)

Theoremdyaddisj 19490* Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadmax 19492* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadmbl 19494* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbllem 19495* Lemma for opnmbl 19496. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbl 19496 All open sets are measurable. This proof, via dyadmbl 19494 and uniioombl 19483, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.)

TheoremopnmblALT 19497 All open sets are measurable. This alternative proof of opnmbl 19496 is significantly shorter, at the expense of invoking countable choice ax-cc 8317. (This was also the original proof before the current opnmbl 19496 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)

Theoremsubopnmbl 19498 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
t

Theoremvolsup2 19499* The volume of is the supremum of the sequence of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremvolcn 19500* The function formed by restricting a measurable set to a closed interval with a varying endpoint produces an increasing continuous function on the reals. (Contributed by Mario Carneiro, 30-Aug-2014.)

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