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

Theoremitg1climres 18901* 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 18902* Lemma for mbfi1fseq 18908. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem2 18903* Lemma for mbfi1fseq 18908. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1fseqlem3 18904* Lemma for mbfi1fseq 18908. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem4 18905* Lemma for mbfi1fseq 18908. 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 18906* Lemma for mbfi1fseq 18908. Verify that describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem6 18907* Lemma for mbfi1fseq 18908. 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 18908* 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 18909* Lemma for mbfi1flim 18910. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfi1flim 18910* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfmullem2 18911* Lemma for mbfmul 18913. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                                                 MblFn

Theoremmbfmullem 18912 Lemma for mbfmul 18913. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                     MblFn

Theoremmbfmul 18913 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn       MblFn

Theoremitg2lcl 18914* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2val 18915* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2l 18916* Elementhood in the set of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lr 18917* Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremxrge0f 18918 A real function is a nonnegative extended real function if all its values are greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)

Theoremitg2cl 18919 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2ub 18920 The integral of a nonnegative real function is an upper bound on the integrals of all simple functions dominated by . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2leub 18921* Any upper bound on the integrals of all simple functions dominated by is greater than , the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2ge0 18922 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2itg1 18923 The integral of a nonnegative simple function using is the same as its value under . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg20 18924 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lecl 18925 If an integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2le 18926 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2const 18927* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitg2const2 18928* When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremitg2seq 18929* Definitional property of the integral: for any function there is a countable sequence of simple functions less than whose integrals converge to the integral of . (This theorem is for the most part unnecessary in lieu of itg2i1fseq 18942, but unlike that theorem this one doesn't require to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitg2uba 18930* Approximate version of itg2ub 18920. If approximately dominates , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2lea 18931* Approximate version of itg2le 18926. If for almost all , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2eqa 18932* Approximate equality of integrals. If for almost all , then . (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitg2mulclem 18933 Lemma for itg2mulc 18934. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremitg2mulc 18934 The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitg2splitlem 18935* Lemma for itg2split 18936. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2split 18936* The integral splits under an almost disjoint union. (The proof avoids the use of itg2add 18946 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2monolem1 18937* Lemma for itg2mono 18940. We show that for any constant less than one, is less than , and so , which is one half of the equality in itg2mono 18940. Consider the sequence . This is an increasing sequence of measurable sets whose union is , and so has an integral which equals in the limit, by itg1climres 18901. Then by taking the limit in , we get as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremitg2monolem2 18938* Lemma for itg2mono 18940. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2monolem3 18939* Lemma for itg2mono 18940. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2mono 18940* The Monotone Convergence Theorem for nonnegative functions. If is a monotone increasing sequence of positive, measurable, real-valued functions, and is the pointwise limit of the sequence, then is the limit of the sequence . (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2i1fseqle 18941* Subject to the conditions coming from mbfi1fseq 18908, the sequence of simple functions are all less than the target function . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq 18942* Subject to the conditions coming from mbfi1fseq 18908, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq2 18943* In an extension to the results of itg2i1fseq 18942, if there is an upper bound on the integrals of the simple functions approaching , then is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq3 18944* Special case of itg2i1fseq2 18943: if the integral of is a real number, then the standard limit relation holds on the integrals of simple functions approaching . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

MblFn                     MblFn

Theoremitg2add 18946 The integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn                     MblFn

Theoremitg2gt0 18947* If the function is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem1 18948* Lemma for itgcn 19029. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem2 18949* Lemma for itgcn 19029. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremitg2cn 18950* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19216 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
MblFn

Theoremibllem 18951 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremisibl 18952* The predicate " is integrable". The "integrable" predicate corresponds roughly to the range of validity of , which is to say that the expression doesn't make sense unless . (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremisibl2 18953* The predicate " is integrable" when is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblmbf 18954 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
MblFn

Theoremiblitg 18955* If a function is integrable, then the integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremdfitg 18956* Evaluate the class substitution in df-itg 18811. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgex 18957 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1f 18958 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1 18959* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg1 18960 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg 18961* Bound-variable hypothesis builder for an integral: if is (effectively) not free in and , it is not free in . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitg 18962* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitgv 18963* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq2 18964 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgresr 18965 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitg0 18966 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgz 18967 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgeq2dv 18968* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremitgmpt 18969* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgcl 18970* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgvallem 18971* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgvallem3 18972* Lemma for itgposval 18982 and itgreval 18983. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremibl0 18973 The zero function is integrable on any measurable set. (Unlike iblconst 19004, this does not require to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremiblcnlem1 18974* Lemma for iblcnlem 18975. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblcnlem 18975* Expand out the forall in isibl2 18953. (Contributed by Mario Carneiro, 6-Aug-2014.)
MblFn

Theoremitgcnlem 18976* Expand out the sum in dfitg 18956. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblrelem 18977* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblposlem 18978* Lemma for iblpos 18979. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblpos 18979* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblre 18980* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgrevallem1 18981* Lemma for itgposval 18982 and itgreval 18983. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgposval 18982* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgreval 18983* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremitgrecl 18984* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremiblcn 18985* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgcnval 18986* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgre 18987* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitgim 18988* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremiblneg 18989* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgneg 18990* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblss 18991* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremiblss2 18992* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgitg2 18993* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fibl 18994 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgitg1 18995* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgle 18996* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgge0 18997* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgss 18998* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgss2 18999* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgeqa 19000* Approximate equality of integrals. If for almost all , then and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)

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