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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremivthlem1 19301* Lemma for ivth 19304. The set of all values with less than is lower bounded by and upper bounded by . (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremivthlem2 19302* Lemma for ivth 19304. Show that the supremum of cannot be less than . If it was, continuity of implies that there are points just above the supremum that are also less than , a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremivthlem3 19303* Lemma for ivth 19304, the intermediate value theorem. Show that cannot be greater than , and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)

Theoremivth 19304* The intermediate value theorem, increasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)

Theoremivth2 19305* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremivthle 19306* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremivthle2 19307* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremivthicc 19308* The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevthicc 19309* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevthicc2 19310* Combine ivthicc 19308 with evthicc 19309 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremcniccbdd 19311* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)

12.2  Integrals

12.2.1  Lebesgue measure

Syntaxcovol 19312 Extend class notation with the outer Lebesgue measure.

Syntaxcvol 19313 Extend class notation with the Lebesgue measure.

Definitiondf-ovol 19314* Define the outer Lebesgue measure for subsets of the reals. Here is a function from the natural numbers 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.)

Definitiondf-vol 19315* 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.)

Theoremovolfcl 19316 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolfioo 19317* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolficc 19318* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolficcss 19319 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovolfsval 19320 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolfsf 19321 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolsf 19322 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolval 19323* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremelovolm 19324* Elementhood in the set of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremelovolmr 19325* Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolmge0 19326* The set is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolcl 19327 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovollb 19328 The outer volume is a lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremovolgelb 19329* The outer volume is the greatest lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremovolge0 19330 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolf 19331 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovollecl 19332 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremovolsslem 19333* Lemma for ovolss 19334. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolss 19334 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolsscl 19335 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremovolssnul 19336 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremovollb2lem 19337* Lemma for ovollb2 19338. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovollb2 19338 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 19328). (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovolctb 19339 The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Theoremovolq 19340 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovolctb2 19341 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovol0 19342 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovolfi 19343 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremovolsn 19344 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)

Theoremovolunlem1a 19345* Lemma for ovolun 19348. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremovolunlem1 19346* Lemma for ovolun 19348. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolunlem2 19347 Lemma for ovolun 19348. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolun 19348 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 19354, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolunnul 19349 Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremovolfiniun 19350* The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremovoliunlem1 19351* Lemma for ovoliun 19354. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliunlem2 19352* Lemma for ovoliun 19354. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliunlem3 19353* Lemma for ovoliun 19354. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliun 19354* 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 19334, 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 19355* 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 19354.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovoliunnul 19356* A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremshft2rab 19357* 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 19358* Lemma for ovolshft 19360. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolshftlem2 19359* Lemma for ovolshft 19360. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolshft 19360* The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremsca2rab 19361* If is a scale of by , then is a scale of by . (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolscalem1 19362* Lemma for ovolsca 19364. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremovolscalem2 19363* Lemma for ovolshft 19360. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theoremovolsca 19364* The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremovolicc1 19365* 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 19366* Lemma for ovolicc2 19371. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem2 19367* Lemma for ovolicc2 19371. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem3 19368* Lemma for ovolicc2 19371. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem4 19369* Lemma for ovolicc2 19371. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2lem5 19370* Lemma for ovolicc2 19371. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc2 19371* The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicc 19372 The measure of a closed interval. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolicopnf 19373 The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremovolre 19374 The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremismbl 19375* 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 19376* From ovolun 19348, 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 19377 A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremvolf 19378 The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremmblvol 19379 The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblss 19380 A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremmblsplit 19381 The defining property of measurability. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremcmmbl 19382 The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl 19383 A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremnulmbl2 19384* 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 19385 A union of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremshftmbl 19386* A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)

Theorem0mbl 19387 The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremrembl 19388 The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoreminmbl 19389 An intersection of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremdifmbl 19390 A difference of measurable sets is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremfiniunmbl 19391* A finite union of measurable sets is measurable. (Contributed by Mario Carneiro, 20-Mar-2014.)

Theoremvolun 19392 The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremvolinun 19393 Addition of non-disjoint sets. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremvolfiniun 19394* 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 19395* Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
..^

Theoremiundisj2 19396* A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Disj ..^

Theoremvoliunlem1 19397* Lemma for voliun 19401. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem2 19398* Lemma for voliun 19401. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremvoliunlem3 19399* Lemma for voliun 19401. (Contributed by Mario Carneiro, 20-Mar-2014.)
Disj

Theoremiunmbl 19400 The measurable sets are closed under countable union. (Contributed by Mario Carneiro, 18-Mar-2014.)

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