P. 217 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 21601-21700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	evthicc 21601*	Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)
$/\$

Theorem	evthicc2 21602*	Combine ivthicc 21600 with evthicc 21601 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)


Theorem	cniccbdd 21603*	A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)
$/\$ $/\$

13.2 Integrals

13.2.1 Lebesgue measure

Syntax	covol 21604	Extend class notation with the outer Lebesgue measure.


Syntax	cvol 21605	Extend class notation with the Lebesgue measure.


Definition	df-ovol 21606*	Define the outer Lebesgue measure for subsets of the reals. Here is a function from the positive integers to pairs with , and the outer volume of the set is the infimum over all such functions such that the union of the open intervals covers of the sum of . (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$

Definition	df-vol 21607*	Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as . (Contributed by Mario Carneiro, 17-Mar-2014.)
$\$

Theorem	ovolfcl 21608	Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$ $/\$

Theorem	ovolfioo 21609*	Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$

Theorem	ovolficc 21610*	Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$

Theorem	ovolficcss 21611	Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)


Theorem	ovolfsval 21612	The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$

Theorem	ovolfsf 21613	Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)


Theorem	ovolsf 21614	Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)


Theorem	ovolval 21615*	The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$

Theorem	elovolm 21616*	Elementhood in the set of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$

Theorem	elovolmr 21617*	Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$

Theorem	ovolmge0 21618*	The set is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$

Theorem	ovolcl 21619	The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)


Theorem	ovollb 21620	The outer volume is a lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)
$/\$

Theorem	ovolgelb 21621*	The outer volume is the greatest lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)
$/\$ $/\$ $/\$

Theorem	ovolge0 21622	The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)


Theorem	ovolf 21623	The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)


Theorem	ovollecl 21624	If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $/\$

Theorem	ovolsslem 21625*	Lemma for ovolss 21626. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$ $/\$ $/\$

Theorem	ovolss 21626	The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)
$/\$

Theorem	ovolsscl 21627	If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $/\$

Theorem	ovolssnul 21628	A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)
$/\$ $/\$

Theorem	ovollb2lem 21629*	Lemma for ovollb2 21630. (Contributed by Mario Carneiro, 24-Mar-2015.)


Theorem	ovollb2 21630	It is often more convenient to do calculations with closed coverings rather than open ones; here we show that it makes no difference (compare ovollb 21620). (Contributed by Mario Carneiro, 24-Mar-2015.)
$/\$

Theorem	ovolctb 21631	The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
$/\$

Theorem	ovolq 21632	The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)


Theorem	ovolctb2 21633	The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)
$/\$

Theorem	ovol0 21634	The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)


Theorem	ovolfi 21635	A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)
$/\$

Theorem	ovolsn 21636	A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)


Theorem	ovolunlem1a 21637*	Lemma for ovolun 21640. (Contributed by Mario Carneiro, 7-May-2015.)
$/\$ $/\$ $/\$

Theorem	ovolunlem1 21638*	Lemma for ovolun 21640. (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$

Theorem	ovolunlem2 21639	Lemma for ovolun 21640. (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$

Theorem	ovolun 21640	The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 21646, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$ $/\$

Theorem	ovolunnul 21641	Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
$/\$ $/\$

Theorem	ovolfiniun 21642*	The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)
$/\$ $/\$

Theorem	ovoliunlem1 21643*	Lemma for ovoliun 21646. (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	ovoliunlem2 21644*	Lemma for ovoliun 21646. (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	ovoliunlem3 21645*	Lemma for ovoliun 21646. (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$

Theorem	ovoliun 21646*	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 21626, 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.)
$/\$ $/\$

Theorem	ovoliun2 21647*	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 21646.) (Contributed by Mario Carneiro, 12-Jun-2014.)
$/\$ $/\$

Theorem	ovoliunnul 21648*	A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
$/\$ $/\$

Theorem	shft2rab 21649*	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.)


Theorem	ovolshftlem1 21650*	Lemma for ovolshft 21652. (Contributed by Mario Carneiro, 22-Mar-2014.)
$/\$

Theorem	ovolshftlem2 21651*	Lemma for ovolshft 21652. (Contributed by Mario Carneiro, 22-Mar-2014.)
$/\$ $/\$

Theorem	ovolshft 21652*	The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)


Theorem	sca2rab 21653*	If is a scale of by , then is a scale of by . (Contributed by Mario Carneiro, 22-Mar-2014.)


Theorem	ovolscalem1 21654*	Lemma for ovolsca 21656. (Contributed by Mario Carneiro, 6-Apr-2015.)


Theorem	ovolscalem2 21655*	Lemma for ovolshft 21652. (Contributed by Mario Carneiro, 22-Mar-2014.)


Theorem	ovolsca 21656*	The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)


Theorem	ovolicc1 21657*	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.)


Theorem	ovolicc2lem1 21658*	Lemma for ovolicc2 21663. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	ovolicc2lem2 21659*	Lemma for ovolicc2 21663. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	ovolicc2lem3 21660*	Lemma for ovolicc2 21663. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	ovolicc2lem4 21661*	Lemma for ovolicc2 21663. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$

Theorem	ovolicc2lem5 21662*	Lemma for ovolicc2 21663. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$

Theorem	ovolicc2 21663*	The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$

Theorem	ovolicc 21664	The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$

Theorem	ovolicopnf 21665	The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)


Theorem	ovolre 21666	The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)


Theorem	ismbl 21667*	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.)
$/\$ $\$

Theorem	ismbl2 21668*	From ovolun 21640, 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.)
$/\$ $\$

Theorem	volres 21669	A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)


Theorem	volf 21670	The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)


Theorem	mblvol 21671	The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)


Theorem	mblss 21672	A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)


Theorem	mblsplit 21673	The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)
$/\$ $/\$ $\$

Theorem	volss 21674	The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
$/\$ $/\$

Theorem	cmmbl 21675	The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$\$

Theorem	nulmbl 21676	A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	nulmbl2 21677*	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.)
$/\$

Theorem	unmbl 21678	A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	shftmbl 21679*	A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
$/\$

Theorem	0mbl 21680	The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	rembl 21681	The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	inmbl 21682	An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	difmbl 21683	A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $\$

Theorem	finiunmbl 21684*	A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
$/\$

Theorem	volun 21685	The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	volinun 21686	Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)
$/\$ $/\$ $/\$

Theorem	volfiniun 21687*	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

Theorem	iundisj 21688*	Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
$\$ ..^

Theorem	iundisj2 21689*	A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj $\$ ..^

Theorem	voliunlem1 21690*	Lemma for voliun 21694. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj $/\$ $\$

Theorem	voliunlem2 21691*	Lemma for voliun 21694. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theorem	voliunlem3 21692*	Lemma for voliun 21694. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theorem	iunmbl 21693	The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	voliun 21694	The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
$/\$ $/\$ Disj

Theorem	volsuplem 21695*	Lemma for volsup 21696. (Contributed by Mario Carneiro, 4-Jul-2014.)
$/\$ $/\$

Theorem	volsup 21696*	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.)
$/\$

Theorem	iunmbl2 21697*	The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	ioombl1lem1 21698*	Lemma for ioombl1 21702. (Contributed by Mario Carneiro, 18-Aug-2014.)
$/\$

Theorem	ioombl1lem2 21699*	Lemma for ioombl1 21702. (Contributed by Mario Carneiro, 18-Aug-2014.)


Theorem	ioombl1lem3 21700*	Lemma for ioombl1 21702. (Contributed by Mario Carneiro, 18-Aug-2014.)
$/\$

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