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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremioovolcl 22601 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremovolfs2 22602 Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorcl2 22603 An open interval with finite volume has real endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremioorf 22604 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf

Theoremioorval 22605* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
inf        inf

Theoremioorinv2 22606* 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

Theoremioorinv 22607* 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

Theoremioorcl 22608* 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

TheoremioorfOLD 22609 Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorf 22604 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremioorvalOLD 22610* Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorval 22605 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremioorinv2OLD 22611* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorinv2 22606 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremioorinvOLD 22612* The function is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorinv 22607 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremioorclOLD 22613* The function does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorcl 22608 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremuniiccdif 22614 A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremuniioovol 22615* 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 22586.) Lemma 565Ca of [Fremlin5] p. 213. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccvol 22616* 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 22586.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem1 22617* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremuniioombllem2a 22618* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 7-May-2015.)
Disj

Theoremuniioombllem2 22619* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by AV, 13-Sep-2020.)
Disj                                                                       inf

Theoremuniioombllem2OLD 22620* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) Obsolete version of uniioombllem2 22619 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Disj

Theoremuniioombllem3a 22621* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 8-May-2015.)
Disj

Theoremuniioombllem3 22622* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem4 22623* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombllem5 22624* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 25-Aug-2014.)
Disj

Theoremuniioombllem6 22625* Lemma for uniioombl 22626. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniioombl 22626* A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 22585.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
Disj

Theoremuniiccmbl 22627* An almost-disjoint union of closed intervals is measurable. (This proof does not use countable choice, unlike iunmbl 22585.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Disj

Theoremdyadf 22628* The function returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadval 22629* Value of the dyadic rational function . (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadovol 22630* Volume of a dyadic rational interval. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadss 22631* 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 22633* 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 22635* Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremdyadmbl 22637* Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbllem 22638* Lemma for opnmbl 22639. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremopnmbl 22639 All open sets are measurable. This proof, via dyadmbl 22637 and uniioombl 22626, 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 22640 All open sets are measurable. This alternative proof of opnmbl 22639 is significantly shorter, at the expense of invoking countable choice ax-cc 8883. (This was also the original proof before the current opnmbl 22639 was discovered.) (Contributed by Mario Carneiro, 17-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsubopnmbl 22641 Sets which are open in a measurable subspace are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
t

Theoremvolsup2 22642* The volume of is the supremum of the sequence of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremvolcn 22643* 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 22644* 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 22645* Lemma for vitali 22650. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem2 22646* Lemma for vitali 22650. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem3 22647* Lemma for vitali 22650. (Contributed by Mario Carneiro, 16-Jun-2014.)
Disj

Theoremvitalilem4 22648* Lemma for vitali 22650. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem5 22649* Lemma for vitali 22650. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitali 22650 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

Syntaxcmbf 22651 Extend class notation with the class of measurable functions.
MblFn

Syntaxcitg1 22652 Extend class notation with the Lebesgue integral for simple functions.

Syntaxcitg2 22653 Extend class notation with the Lebesgue integral for nonnegative functions.

Syntaxcibl 22654 Extend class notation with the class of integrable functions.

Syntaxcitg 22655 Extend class notation with the general Lebesgue integral.

Definitiondf-mbf 22656* 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 22558) 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 22657* 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 22658* 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 22659* 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 22660* 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 22658 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 22658 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremismbf1 22661* The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 22665 and ismbfcn 22666 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbff 22662 A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfdm 22663 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfconstlem 22664 Lemma for mbfconst 22670. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremismbf 22665* 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 22558. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremismbfcn 22666 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

Theoremmbfima 22667 Definitional property of a measurable function: the preimage of an open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfimaicc 22668 The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimasn 22669 The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfconst 22670 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfid 22671 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfmptcl 22672* Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn

Theoremmbfdm2 22673* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremismbfcn2 22674* A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.)
MblFn MblFn MblFn

Theoremismbfd 22675* Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 22689. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremismbf2d 22676* Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfeqalem 22677* Lemma for mbfeqa 22678. (Contributed by Mario Carneiro, 2-Sep-2014.)
MblFn MblFn

Theoremmbfeqa 22678* If two functions are equal almost everywhere, then one is measurable iff the other is. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
MblFn MblFn

Theoremmbfres 22679 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn MblFn

Theoremmbfres2 22680 Measurability of a piecewise function: if is measurable on subsets and of its domain, and these pieces make up all of , then is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn       MblFn              MblFn

Theoremmbfss 22681* Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfmulc2lem 22682 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn                     MblFn

Theoremmbfmulc2re 22683 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn                     MblFn

Theoremmbfmax 22684* The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn              MblFn              MblFn

Theoremmbfneg 22685* The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfpos 22686* The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfposr 22687* Converse to mbfpos 22686. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfposb 22688* A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn MblFn MblFn

Theoremismbf3d 22689* Simplified form of ismbfd 22675. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimaopnlem 22690* Lemma for mbfimaopn 22691. (Contributed by Mario Carneiro, 25-Aug-2014.)
fld                            MblFn

Theoremmbfimaopn 22691 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 22693, which explains why is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
fld       MblFn

Theoremmbfimaopn2 22692 The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
fld       t        MblFn

Theoremcncombf 22693 The composition of a continuous function with a measurable function is measurable. (More generally, can be a Borel-measurable function, but notably the condition that be only measurable is too weak, the usual counterexample taking to be the Cantor function and the indicator function of the -image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn MblFn

Theoremcnmbf 22694 A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
MblFn

Theoremmbfaddlem 22695 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn                     MblFn

Theoremmbfadd 22696 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfsub 22697 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn       MblFn       MblFn

Theoremmbfmulc2 22698* A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfsup 22699* The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, is a function of both and , since it is an -indexed sequence of functions on . (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
MblFn                     MblFn

Theoremmbfinf 22700* The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)
inf               MblFn                     MblFn

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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