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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremismbl 18901* 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 18902* From ovolun 18874, 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 18903 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremvolf 18904 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremmblvol 18905 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblss 18906 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblsplit 18907 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremcmmbl 18908 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl 18909 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl2 18910* 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 18911 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremshftmbl 18912* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theorem0mbl 18913 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremrembl 18914 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoreminmbl 18915 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremdifmbl 18916 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremfiniunmbl 18917* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)

Theoremvolun 18918 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremvolinun 18919 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremvolfiniun 18920* 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 18921* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
..^

Theoremiundisj2 18922* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj ..^

Theoremvoliunlem1 18923* Lemma for voliun 18927. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem2 18924* Lemma for voliun 18927. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem3 18925* Lemma for voliun 18927. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremiunmbl 18926 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremvoliun 18927 The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Disj

Theoremvolsuplem 18928* Lemma for volsup 18929. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremvolsup 18929* 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 18930* The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremioombl1lem1 18931* Lemma for ioombl1 18935. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem2 18932* Lemma for ioombl1 18935. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem3 18933* Lemma for ioombl1 18935. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremioombl1lem4 18934* Lemma for ioombl1 18935. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremioombl1 18935 An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Theoremicombl1 18936 A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremicombl 18937 A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremioombl 18938 An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremiccmbl 18939 A closed real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremiccvolcl 18940 A closed real interval has finite volume. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremovolioo 18941 The measure of an open interval. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremovolfs2 18942 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorcl2 18943 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorf 18944 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorval 18945* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorinv2 18946* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorinv 18947* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorcl 18948* The function does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremuniiccdif 18949 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremuniioovol 18950* 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 18927.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccvol 18951* 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 18927.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem1 18952* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem2a 18953* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 7-May-2015.)
Disj

Theoremuniioombllem2 18954* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj

Theoremuniioombllem3a 18955* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 8-May-2015.)
Disj

Theoremuniioombllem3 18956* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem4 18957* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem5 18958* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 25-Aug-2014.)
Disj

Theoremuniioombllem6 18959* Lemma for uniioombl 18960. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombl 18960* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18926.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccmbl 18961* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 18926.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremdyadf 18962* The function returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadval 18963* Value of the dyadic rational function . (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadovol 18964* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadss 18965* 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 18967* 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 18969* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadmbl 18971* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbllem 18972* Lemma for opnmbl 18973. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbl 18973 All open sets are measurable. This proof, via dyadmbl 18971 and uniioombl 18960, 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 18974 All open sets are measurable. This alternative proof of opnmbl 18973 is significantly shorter, at the expense of invoking countable choice ax-cc 8077. (This was also the original proof before the current opnmbl 18973 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (Proof modification is discouraged.)

Theoremsubopnmbl 18975 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
t

Theoremvolsup2 18976* The volume of is the supremum of the sequence of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremvolcn 18977* 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.)

Theoremvolivth 18978* 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.)

Theoremvitalilem1 18979* Lemma for vitali 18984. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem2 18980* Lemma for vitali 18984. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem3 18981* Lemma for vitali 18984. (Contributed by Mario Carneiro, 16-Jun-2014.)
Disj

Theoremvitalilem4 18982* Lemma for vitali 18984. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem5 18983* Lemma for vitali 18984. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitali 18984 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.)

12.2.2  Lebesgue integration

Syntaxcmbf 18985 Extend class notation with the class of measurable functions.
MblFn

Syntaxcitg1 18986 Extend class notation with the Lebesgue integral for simple functions.

Syntaxcitg2 18987 Extend class notation with the Lebesgue integral for nonnegative functions.

Syntaxcibl 18988 Extend class notation with the class of integrable functions.

Syntaxcitg 18989 Extend class notation with the general Lebesgue integral.

Syntaxcdit 18990 Extend class notation with the directed integral.
_

Definitiondf-mbf 18991* 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 18901) 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

Definitiondf-itg1 18992* 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

Definitiondf-itg2 18993* 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.)

Definitiondf-ibl 18994* 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

Definitiondf-itg 18995* 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 18993 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 18993 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)

Definitiondf-ditg 18996 Define the directed integral, which is just a regular integral but with a sign change when the limits are interchanged. The and here are the lower and upper limits of the integral, usually written as a subscript and superscript next to the integral sign. We define the region of integration to be an open interval instead of closed so that we can use for limits and also integrate up to a singularity at an endpoint. (Contributed by Mario Carneiro, 13-Aug-2014.)
_

Theoremismbf1 18997* The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19001 and ismbfcn 19002 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbff 18998 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfdm 18999 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfconstlem 19000 Lemma for mbfconst 19006. (Contributed by Mario Carneiro, 17-Jun-2014.)

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