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

Theoremismbf1 19501* The predicate " is a measurable function". This is more naturally stated for functions on the reals, see ismbf 19505 and ismbfcn 19506 for the decomposition of the real and imaginary parts. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbff 19502 A measurable function is a function into the complexes. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfdm 19503 The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfconstlem 19504 Lemma for mbfconst 19510. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremismbf 19505* 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 19405. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremismbfcn 19506 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 19507 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 19508 The preimage of any closed interval under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimasn 19509 The preimage of a point under a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfconst 19510 A constant function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfid 19511 The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.)
MblFn

Theoremmbfmptcl 19512* Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn

Theoremmbfdm2 19513* The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremismbfcn2 19514* 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 19515* 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 19529. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremismbf2d 19516* Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfeqalem 19517* Lemma for mbfeqa 19518. (Contributed by Mario Carneiro, 2-Sep-2014.)
MblFn MblFn

Theoremmbfeqa 19518* 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 19519 The restriction of a measurable function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn MblFn

Theoremmbfres2 19520 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 19521* Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfmulc2lem 19522 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn                     MblFn

Theoremmbfmulc2re 19523 Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn                     MblFn

Theoremmbfmax 19524* The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn              MblFn              MblFn

Theoremmbfneg 19525* The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfpos 19526* The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
MblFn       MblFn

Theoremmbfposr 19527* Converse to mbfpos 19526. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfposb 19528* A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
MblFn MblFn MblFn

Theoremismbf3d 19529* Simplified form of ismbfd 19515. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremmbfimaopnlem 19530* Lemma for mbfimaopn 19531. (Contributed by Mario Carneiro, 25-Aug-2014.)
fld                            MblFn

Theoremmbfimaopn 19531 The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 19533, which explains why is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
fld       MblFn

Theoremmbfimaopn2 19532 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 19533 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 19534 A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
MblFn

Theoremmbfaddlem 19535 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn                     MblFn

Theoremmbfadd 19536 The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
MblFn       MblFn       MblFn

Theoremmbfsub 19537 The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn       MblFn       MblFn

Theoremmbfmulc2 19538* A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn       MblFn

Theoremmbfsup 19539* 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 19540* The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn                     MblFn

Theoremmbflimsup 19541* 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 19542* The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn              MblFn

Theoremmbflim 19543* The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn              MblFn

Syntaxc0p 19544 Extend class notation to include the zero polynomial.

Definitiondf-0p 19545 Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theorem0pval 19546 The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theorem0plef 19547 Two ways to say that the function on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theorem0pledm 19548 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 19549 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 19499); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremi1fmbf 19550 Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
MblFn

Theoremi1ff 19551 A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1frn 19552 A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima 19553 Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima2 19554 Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fima2sn 19555 Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1fd 19556* A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremi1f0rn 19557 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 19558* The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg1val2 19559* The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1cl 19560 Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1ge0 19561 Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1f0 19562 The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg10 19563 The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1f1lem 19564* Lemma for i1f1 19565 and itg11 19566. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1f1 19565* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremitg11 19566* 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 19567* 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 19568* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1fmullem 19569* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremi1fadd 19570 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremi1fmul 19571 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)

Theoremitg1addlem2 19572* Lemma for itg1add 19576. 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 19574 and itg1addlem5 19575. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremitg1add 19576 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremi1fmulclem 19577 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremi1fmulc 19578 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremitg1mulc 19579 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremi1fres 19580* 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 19581* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremi1fposd 19582* Deduction form of i1fposd 19582. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fsub 19583 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitg1sub 19584 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitg10a 19585* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg1ge0a 19586* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg1lea 19587* Approximate version of itg1le 19588. If for almost all , then . (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)

Theoremitg1le 19588 If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)

Theoremitg1climres 19589* Restricting the simple function to the increasing sequence of measurable sets whose union is yields a sequence of simple functions whose integrals approach the integral of . (Contributed by Mario Carneiro, 15-Aug-2014.)

Theoremmbfi1fseqlem1 19590* Lemma for mbfi1fseq 19596. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem2 19591* Lemma for mbfi1fseq 19596. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1fseqlem3 19592* Lemma for mbfi1fseq 19596. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem4 19593* Lemma for mbfi1fseq 19596. This lemma is not as interesting as it is long - it is simply checking that is in fact a sequence of simple functions, by verifying that its range is in (which is to say, the numbers from to in increments of ), and also that the preimage of each point is measurable, because it is equal to for and for . (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem5 19594* Lemma for mbfi1fseq 19596. Verify that describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem6 19595* Lemma for mbfi1fseq 19596. Verify that converges pointwise to , and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1fseq 19596* A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1flimlem 19597* Lemma for mbfi1flim 19598. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfi1flim 19598* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfmullem2 19599* Lemma for mbfmul 19601. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                                                 MblFn

Theoremmbfmullem 19600 Lemma for mbfmul 19601. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                     MblFn

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