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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvitalilem2 19501* Lemma for vitali 19505. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem3 19502* Lemma for vitali 19505. (Contributed by Mario Carneiro, 16-Jun-2014.)
Disj

Theoremvitalilem4 19503* Lemma for vitali 19505. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitalilem5 19504* Lemma for vitali 19505. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremvitali 19505 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 19506 Extend class notation with the class of measurable functions.
MblFn

Syntaxcitg1 19507 Extend class notation with the Lebesgue integral for simple functions.

Syntaxcitg2 19508 Extend class notation with the Lebesgue integral for nonnegative functions.

Syntaxcibl 19509 Extend class notation with the class of integrable functions.

Syntaxcitg 19510 Extend class notation with the general Lebesgue integral.

Syntaxcdit 19511 Extend class notation with the directed integral.
_

Definitiondf-mbf 19512* 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 19422) 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 19513* 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 19514* 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 19515* 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 19516* 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 19514 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 19514 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)

Definitiondf-ditg 19517 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 19518* The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19522 and ismbfcn 19523 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbff 19519 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfdm 19520 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfconstlem 19521 Lemma for mbfconst 19527. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremismbf 19522* 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 19422. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremismbfcn 19523 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 19524 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 19525 The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimasn 19526 The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfconst 19527 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfid 19528 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfmptcl 19529* Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn

Theoremmbfdm2 19530* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremismbfcn2 19531* 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 19532* 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 19546. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremismbf2d 19533* Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfeqalem 19534* Lemma for mbfeqa 19535. (Contributed by Mario Carneiro, 2-Sep-2014.)
MblFn MblFn

Theoremmbfeqa 19535* 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 19536 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn MblFn

Theoremmbfres2 19537 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 19538* Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfmulc2lem 19539 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn                     MblFn

Theoremmbfmulc2re 19540 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn                     MblFn

Theoremmbfmax 19541* The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn              MblFn              MblFn

Theoremmbfneg 19542* The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfpos 19543* The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfposr 19544* Converse to mbfpos 19543. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfposb 19545* A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn MblFn MblFn

Theoremismbf3d 19546* Simplified form of ismbfd 19532. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimaopnlem 19547* Lemma for mbfimaopn 19548. (Contributed by Mario Carneiro, 25-Aug-2014.)
fld                            MblFn

Theoremmbfimaopn 19548 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 19550, which explains why is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
fld       MblFn

Theoremmbfimaopn2 19549 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 19550 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 19551 A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
MblFn

Theoremmbfaddlem 19552 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn                     MblFn

Theoremmbfadd 19553 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfsub 19554 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn       MblFn       MblFn

Theoremmbfmulc2 19555* A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfsup 19556* 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 19557* The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn                     MblFn

Theoremmbflimsup 19558* The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 9-May-2016.)
MblFn              MblFn

Theoremmbflimlem 19559* The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn              MblFn

Theoremmbflim 19560* The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn              MblFn

Syntaxc0p 19561 Extend class notation to include the zero polynomial.

Definitiondf-0p 19562 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theorem0pval 19563 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theorem0plef 19564 Two ways to say that the function on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theorem0pledm 19565 Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremisi1f 19566 The predicate " is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom to represent this concept because is the first preparation function for our final definition (see df-itg 19516); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremi1fmbf 19567 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremi1ff 19568 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1frn 19569 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima 19570 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima2 19571 Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima2sn 19572 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fd 19573* A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1f0rn 19574 Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg1val 19575* The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg1val2 19576* The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1cl 19577 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1ge0 19578 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1f0 19579 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg10 19580 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1f1lem 19581* Lemma for i1f1 19582 and itg11 19583. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1f1 19582* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg11 19583* The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitg1addlem1 19584* Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremi1faddlem 19585* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1fmullem 19586* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1fadd 19587 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1fmul 19588 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)

Theoremitg1addlem2 19589* Lemma for itg1add 19593. The function represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both and are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 19591 and itg1addlem5 19592. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1addlem3 19590* Lemma for itg1add 19593. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1addlem4 19591* Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg1addlem5 19592* Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)

Theoremitg1add 19593 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremi1fmulclem 19594 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremi1fmulc 19595 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremitg1mulc 19596 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremi1fres 19597* The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside .) (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremi1fpos 19598* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremi1fposd 19599* Deduction form of i1fposd 19599. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fsub 19600 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)

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