P. 234 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 23301-23400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	volss 23301	The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017.)
$/\$ $/\$

Theorem	cmmbl 23302	The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$\$

Theorem	nulmbl 23303	A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	nulmbl2 23304*	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 23305	A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	shftmbl 23306*	A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
$/\$

Theorem	0mbl 23307	The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	rembl 23308	The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	unidmvol 23309	The union of the Lebesgue measurable sets is . (Contributed by Thierry Arnoux, 30-Jan-2017.)


Theorem	inmbl 23310	An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	difmbl 23311	A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $\$

Theorem	finiunmbl 23312*	A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)
$/\$

Theorem	volun 23313	The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	volinun 23314	Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)
$/\$ $/\$ $/\$

Theorem	volfiniun 23315*	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 23316*	Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
$\$ ..^

Theorem	iundisj2 23317*	A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj $\$ ..^

Theorem	voliunlem1 23318*	Lemma for voliun 23322. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj $/\$ $\$

Theorem	voliunlem2 23319*	Lemma for voliun 23322. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theorem	voliunlem3 23320*	Lemma for voliun 23322. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theorem	iunmbl 23321	The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)


Theorem	voliun 23322	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 23323*	Lemma for volsup 23324. (Contributed by Mario Carneiro, 4-Jul-2014.)
$/\$ $/\$

Theorem	volsup 23324*	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 23325*	The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$

Theorem	ioombl1lem1 23326*	Lemma for ioombl1 23330. (Contributed by Mario Carneiro, 18-Aug-2014.)
$/\$

Theorem	ioombl1lem2 23327*	Lemma for ioombl1 23330. (Contributed by Mario Carneiro, 18-Aug-2014.)


Theorem	ioombl1lem3 23328*	Lemma for ioombl1 23330. (Contributed by Mario Carneiro, 18-Aug-2014.)
$/\$

Theorem	ioombl1lem4 23329*	Lemma for ioombl1 23330. (Contributed by Mario Carneiro, 16-Jun-2014.)
$\$

Theorem	ioombl1 23330	An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)


Theorem	icombl1 23331	A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)


Theorem	icombl 23332	A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$

Theorem	ioombl 23333	An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)


Theorem	iccmbl 23334	A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$

Theorem	iccvolcl 23335	A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)
$/\$

Theorem	ovolioo 23336	The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)
$/\$ $/\$

Theorem	volioo 23337	The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
$/\$ $/\$

Theorem	ioovolcl 23338	An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
$/\$

Theorem	ovolfs2 23339	Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)


Theorem	ioorcl2 23340	An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$ $/\$

Theorem	ioorf 23341	Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf

Theorem	ioorval 23342*	Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf inf

Theorem	ioorinv2 23343*	The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf

Theorem	ioorinv 23344*	The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf

Theorem	ioorcl 23345*	The function does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf $/\$

Theorem	uniiccdif 23346	A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
$/\$ $\$

Theorem	uniioovol 23347*	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 23322.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theorem	uniiccvol 23348*	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 23322.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theorem	uniioombllem1 23349*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theorem	uniioombllem2a 23350*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 7-May-2015.)
Disj $/\$ $/\$

Theorem	uniioombllem2 23351*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by AV, 13-Sep-2020.)
Disj inf $/\$

Theorem	uniioombllem3a 23352*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 8-May-2015.)
Disj $/\$

Theorem	uniioombllem3 23353*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj $\$ $\$

Theorem	uniioombllem4 23354*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theorem	uniioombllem5 23355*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 25-Aug-2014.)
Disj $\$

Theorem	uniioombllem6 23356*	Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj $\$

Theorem	uniioombl 23357*	A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23321.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theorem	uniiccmbl 23358*	An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 23321.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theorem	dyadf 23359*	The function returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)


Theorem	dyadval 23360*	Value of the dyadic rational function . (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$

Theorem	dyadovol 23361*	Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$

Theorem	dyadss 23362*	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.)
$/\$ $/\$ $/\$

Theorem	dyaddisjlem 23363*	Lemma for dyaddisj 23364. (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$ $/\$ $/\$ $/\$ $\/$ $\/$

Theorem	dyaddisj 23364*	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.)
$/\$ $\/$ $\/$

Theorem	dyadmaxlem 23365*	Lemma for dyadmax 23366. (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$

Theorem	dyadmax 23366*	Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
$/\$

Theorem	dyadmbllem 23367*	Lemma for dyadmbl 23368. (Contributed by Mario Carneiro, 26-Mar-2015.)


Theorem	dyadmbl 23368*	Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)


Theorem	opnmbllem 23369*	Lemma for opnmbl 23370. (Contributed by Mario Carneiro, 26-Mar-2015.)


Theorem	opnmbl 23370	All open sets are measurable. This proof, via dyadmbl 23368 and uniioombl 23357, 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.)


Theorem	opnmblALT 23371	All open sets are measurable. This alternative proof of opnmbl 23370 is significantly shorter, at the expense of invoking countable choice ax-cc 9257. (This was also the original proof before the current opnmbl 23370 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)


Theorem	subopnmbl 23372	Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
↾_t $/\$

Theorem	volsup2 23373*	The volume of is the supremum of the sequence of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)
$/\$ $/\$

Theorem	volcn 23374*	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.)
$/\$

Theorem	volivth 23375*	The Intermediate Value Theorem for the Lebesgue volume function. For any positive , there is a measurable subset of whose volume is . (Contributed by Mario Carneiro, 30-Aug-2014.)
$/\$ $/\$

Theorem	vitalilem1 23376*	Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.)
$/\$ $/\$

Theorem	vitalilem1OLD 23377*	Obsolete proof of vitalilem1 23376 as of 1-May-2021. Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$ $/\$

Theorem	vitalilem2 23378*	Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$ $/\$ $\$ $/\$ $/\$

Theorem	vitalilem3 23379*	Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$ $/\$ $\$ Disj

Theorem	vitalilem4 23380*	Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$ $/\$ $\$ $/\$

Theorem	vitalilem5 23381*	Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)
$/\$ $/\$ $\$

Theorem	vitali 23382	If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)


13.2.2 Lebesgue integration

13.2.2.1 Lesbesgue integral

Syntax	cmbf 23383	Extend class notation with the class of measurable functions.
MblFn

Syntax	citg1 23384	Extend class notation with the Lebesgue integral for simple functions.


Syntax	citg2 23385	Extend class notation with the Lebesgue integral for nonnegative functions.


Syntax	cibl 23386	Extend class notation with the class of integrable functions.


Syntax	citg 23387	Extend class notation with the general Lebesgue integral.


Definition	df-mbf 23388*	Define the class of measurable functions on the reals. A real function is measurable if the preimage of every open interval is a measurable set (see ismbl 23294) and a complex function is measurable if the real and imaginary parts of the function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn $/\$

Definition	df-itg1 23389*	Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn $/\$ $/\$ $\$ $\$

Definition	df-itg2 23390*	Define the Lebesgue integral for nonnegative functions. A nonnegative function's integral is the supremum of the integrals of all simple functions that are less than the input function. Note that this may be for functions that take the value on a set of positive measure or functions that are bounded below by a positive number on a set of infinite measure. (Contributed by Mario Carneiro, 28-Jun-2014.)
$/\$

Definition	df-ibl 23391*	Define the class of integrable functions on the reals. A function is integrable if it is measurable and the integrals of the pieces of the function are all finite. (Contributed by Mario Carneiro, 28-Jun-2014.)
MblFn $/\$

Definition	df-itg 23392*	Define the full Lebesgue integral, for complex-valued functions to . The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of from to is . The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 23390 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 23390 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
$/\$

Theorem	ismbf1 23393*	The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 23397 and ismbfcn 23398 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn $/\$ $/\$

Theorem	mbff 23394	A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theorem	mbfdm 23395	The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theorem	mbfconstlem 23396	Lemma for mbfconst 23402. (Contributed by Mario Carneiro, 17-Jun-2014.)
$/\$

Theorem	ismbf 23397*	The predicate " is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 23294. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theorem	ismbfcn 23398	A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn MblFn $/\$ MblFn

Theorem	mbfima 23399	Definitional property of a measurable function: the preimage of an open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn $/\$

Theorem	mbfimaicc 23400	The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn $/\$ $/\$ $/\$

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