:: Product Pre-Measure
:: by Noboru Endou
::
:: Received December 31, 2015
:: Copyright (c) 2015-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, CARD_3, FUNCT_1, RELAT_1, FINSEQ_1, XBOOLE_0,
ARYTM_3, XXREAL_0, CARD_1, TARSKI, NAT_1, ZFMISC_1, ORDINAL4, PROB_1,
FINSUB_1, SETFAM_1, PROB_2, MEASURE9, MSSUBFAM, FUNCOP_1, SUPINF_2,
VALUED_0, FUNCT_2, PARTFUN1, MEASURE1, ORDINAL2, SERIES_1, MESFUNC5,
SUPINF_1, SEQ_2, MEASURE6, SEQFUNC, REAL_1, XCMPLX_0, PBOOLE, MESFUNC9,
VALUED_1, SIMPLEX0, SRINGS_3, MEASURE3, MEASURE5, XXREAL_1, MEASUR10,
SRINGS_1, SRINGS_4, MESFUNC8, MESFUNC1, FUNCT_3, INTEGRA5, RFUNCT_3;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1,
ORDINAL1, RELSET_1, PARTFUN1, FUNCT_2, FUNCOP_1, FINSUB_1, CARD_1,
CARD_3, NUMBERS, FUNCT_3, XXREAL_1, XXREAL_3, BINOP_1, XTUPLE_0, PROB_1,
XCMPLX_0, XREAL_0, XXREAL_0, SIMPLEX0, NAT_1, VALUED_0, SUPINF_1,
FINSEQ_1, FINSEQOP, SUPINF_2, PROB_2, SEQFUNC, MEASURE1, MEASURE3,
MEASURE5, MESFUNC1, MESFUNC2, MESFUNC5, MESFUNC8, MESFUNC9, RINFSUP2,
EXTREAL1, SRINGS_3, MEASURE9, SRINGS_1, SRINGS_4, MESFUNC6;
constructors TOPMETR, PROB_2, MESFUNC5, EXTREAL1, SUPINF_1, MESFUNC1,
MESFUNC9, RINFSUP2, FINSEQOP, SEQFUNC, MEASURE3, INTEGRA1, SRINGS_1,
MEASURE9, SEQ_4, SRINGS_4, MESFUNC2, MESFUNC8, MESFUNC6;
registrations XREAL_0, MEMBERED, FUNCT_1, XBOOLE_0, FINSEQ_1, RELAT_1,
SUBSET_1, XXREAL_0, ROUGHS_1, NUMBERS, VALUED_0, FUNCT_2, ORDINAL1,
MESFUNC9, RELSET_1, PROB_1, MEASURE1, MESFUNC5, NAT_1, PARTFUN1,
XXREAL_3, CARD_1, XXREAL_2, SIMPLEX0, SRINGS_3, DBLSEQ_3, MEASURE3,
MEASURE9, SRINGS_2, SRINGS_1, SRINGS_4;
requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
begin :: Preliminaries
theorem :: MEASUR10:1
for A,A1,A2,B,B1,B2 be non empty set holds
[:A1,B1:] misses [:A2,B2:] & [:A,B:] = [:A1,B1:] \/ [:A2,B2:]
iff
(A1 misses A2 & A = A1 \/ A2 & B = B1 & B = B2)
or (B1 misses B2 & B = B1 \/ B2 & A = A1 & A = A2);
definition
let C,D be non empty set, F be sequence of Funcs(C,D), n be Nat;
redefine func F.n -> Function of C,D;
end;
theorem :: MEASUR10:2
for X,Y,A,B being set, x,y being object st x in X & y in Y holds
chi(A,X).x * chi(B,Y).y = chi([:A,B:],[:X,Y:]).(x,y);
registration
let A, B be set;
cluster chi (A,B) -> nonnegative;
end;
theorem :: MEASUR10:3
:: Construction of complete sigma measure from pre-measure
for X be non empty set, S be semialgebra_of_sets of X,
P be pre-Measure of S, m be induced_Measure of S,P,
M be induced_sigma_Measure of S,m holds
COM(M) is_complete COM(sigma(Field_generated_by S),M);
definition
func Family_of_Intervals -> semialgebra_of_sets of REAL equals
:: MEASUR10:def 1
the set of all I where I is Interval;
end;
theorem :: MEASUR10:4
Family_of_halflines c= Family_of_Intervals;
theorem :: MEASUR10:5
for I be Subset of REAL st I is Interval holds I in Borel_Sets;
theorem :: MEASUR10:6
sigma Family_of_Intervals = Borel_Sets &
sigma(Field_generated_by Family_of_Intervals) = Borel_Sets;
begin :: Family of semialgebras, fields and measures
theorem :: MEASUR10:7
for X1,X2 be set, S1 be non empty Subset-Family of X1,
S2 be non empty Subset-Family of X2
holds the set of all [:a,b:] where a is Element of S1, b is Element of S2
is non empty Subset-Family of [:X1,X2:];
theorem :: MEASUR10:8
for X,Y be set, M be with_empty_element Subset-Family of X,
N be with_empty_element Subset-Family of Y
holds the set of all [:A,B:] where A is Element of M, B is Element of N
is with_empty_element Subset-Family of [:X,Y:];
theorem :: MEASUR10:9
for X be set, F1,F2 be disjoint_valued FinSequence of X
st union rng F1 misses union rng F2 holds
F1^F2 is disjoint_valued FinSequence of X;
theorem :: MEASUR10:10
for X1,X2 be set, S1 be Semiring of X1, S2 be Semiring of X2 holds
the set of all [:A,B:] where A is Element of S1, B is Element of S2
is Semiring of [:X1,X2:];
theorem :: MEASUR10:11
for X1,X2 be set, S1 be semialgebra_of_sets of X1,
S2 be semialgebra_of_sets of X2 holds
the set of all [:A,B:] where A is Element of S1, B is Element of S2
is semialgebra_of_sets of [:X1,X2:];
theorem :: MEASUR10:12
for X1,X2 be set, F1 be Field_Subset of X1, F2 be Field_Subset of X2
holds the set of all [:A,B:] where A is Element of F1, B is Element of F2
is semialgebra_of_sets of [:X1,X2:];
definition
let n be non zero Nat, X be non-empty n-element FinSequence;
mode SemialgebraFamily of X -> n-element FinSequence means
:: MEASUR10:def 2
for i being Nat st i in Seg n holds
it.i is semialgebra_of_sets of X.i;
end;
definition
let n be non zero Nat, X be non-empty n-element FinSequence;
redefine mode SemialgebraFamily of X
-> cap-closed-yielding SemiringFamily of X;
end;
theorem :: MEASUR10:13
for n be non zero Nat, X be non-empty n-element FinSequence,
S be SemialgebraFamily of X
holds for i be Nat st i in Seg n holds X.i in S.i;
theorem :: MEASUR10:14
for X be non-empty 1-element FinSequence,
S be SemialgebraFamily of X holds
the set of all product <*s*> where s is Element of S.1 is
semialgebra_of_sets of the set of all <*x*> where x is Element of X.1;
theorem :: MEASUR10:15
for X be non-empty 1-element FinSequence, S be SemialgebraFamily of X
holds SemiringProduct(S) is semialgebra_of_sets of product X;
theorem :: MEASUR10:16
for X1,X2 be set, S1 be semialgebra_of_sets of X1,
S2 be semialgebra_of_sets of X2 holds
the set of all [:s1,s2:] where s1 is Element of S1,
s2 is Element of S2 is semialgebra_of_sets of [:X1,X2:];
theorem :: MEASUR10:17
for n be non zero Nat, X be non-empty n-element FinSequence,
S be SemialgebraFamily of X holds
SemiringProduct(S) is semialgebra_of_sets of product X;
theorem :: MEASUR10:18
for n be non zero Nat, Xn be non-empty n-element FinSequence,
X1 be non-empty 1-element FinSequence,
Sn be SemialgebraFamily of Xn, S1 be SemialgebraFamily of X1 holds
SemiringProduct(Sn^S1) is semialgebra_of_sets of product(Xn^X1);
definition
let n be non zero Nat, X be non-empty n-element FinSequence;
mode FieldFamily of X -> n-element FinSequence means
:: MEASUR10:def 3
for i being Nat st i in Seg n holds it.i is Field_Subset of X.i;
end;
definition
let n be non zero Nat, X be non-empty n-element FinSequence;
redefine mode FieldFamily of X -> SemialgebraFamily of X;
end;
theorem :: MEASUR10:19
for X be non-empty 1-element FinSequence,
S be FieldFamily of X holds
the set of all product <*s*> where s is Element of S.1 is
Field_Subset of the set of all <*x*> where x is Element of X.1;
theorem :: MEASUR10:20
for X be non-empty 1-element FinSequence, S be FieldFamily of X
holds SemiringProduct(S) is Field_Subset of product X;
definition
let n be non zero Nat, X be non-empty n-element FinSequence,
S be FieldFamily of X;
mode MeasureFamily of S -> n-element FinSequence means
:: MEASUR10:def 4
for i being Nat st i in Seg n
ex F being Field_Subset of X.i st F = S.i & it.i is Measure of F;
end;
begin :: Product of two measures
definition
let X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2;
func measurable_rectangles(S1,S2) -> semialgebra_of_sets of [:X1,X2:]
equals
:: MEASUR10:def 5
the set of all [:A,B:] where A is Element of S1, B is Element of S2;
end;
theorem :: MEASUR10:21
for X being set, F being Field_Subset of X
ex S being semialgebra_of_sets of X
st F = S & F = Field_generated_by S;
definition
let X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2,
m1 be Measure of S1, m2 be Measure of S2;
func product-pre-Measure(m1,m2) -> nonnegative zeroed Function
of measurable_rectangles(S1,S2),ExtREAL means
:: MEASUR10:def 6
for C be Element of measurable_rectangles(S1,S2)
ex A be Element of S1, B be Element of S2 st
C = [:A,B:] & it.C = m1.A * m2.B;
end;
theorem :: MEASUR10:22
for X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2,
m1 be Measure of S1, m2 be Measure of S2, A,B be set
st A in S1 & B in S2 holds
product-pre-Measure(m1,m2).([:A,B:]) = m1.A * m2.B;
theorem :: MEASUR10:23
for X1,X2 be set, S1 be non empty Subset-Family of X1,
S2 be non empty Subset-Family of X2,
S be non empty Subset-Family of [:X1,X2:], H be FinSequence of S
st S = the set of all [:A,B:] where A is Element of S1, B is Element of S2
ex F be FinSequence of S1, G be FinSequence of S2 st
len H = len F & len H = len G &
for k be Nat st k in dom H & H.k <> {} holds H.k = [:F.k,G.k:];
theorem :: MEASUR10:24
for X be set,
S be non empty semi-diff-closed cap-closed Subset-Family of X,
E1,E2 be Element of S
ex F1,F2,F3 be disjoint_valued FinSequence of S
st union rng F1 = E1 \ E2
& union rng F2 = E2 \ E1 & union rng F3 = E1 /\ E2
& F1^F2^F3 is disjoint_valued FinSequence of S;
theorem :: MEASUR10:25
for X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2,
m1 be Measure of S1, m2 be Measure of S2,
E1,E2 be Element of measurable_rectangles(S1,S2)
st E1 misses E2 & E1 \/ E2 in measurable_rectangles(S1,S2)
holds product-pre-Measure(m1,m2).(E1 \/ E2)
= product-pre-Measure(m1,m2).E1 + product-pre-Measure(m1,m2).E2;
theorem :: MEASUR10:26
for X be non empty set, S be non empty Subset-Family of X,
f be Function of NAT,S, F be Functional_Sequence of X,ExtREAL st
f is disjoint_valued & (for n be Nat holds F.n = chi(f.n,X))
holds (for x be object st x in X holds
chi(Union f,X).x = (lim Partial_Sums F).x);
theorem :: MEASUR10:27
for X being non empty set,
S being SigmaField of X, M being sigma_Measure of S,
f being PartFunc of X,ExtREAL, r being Real
st dom f in S & 0 <= r & (for x be object st x in dom f holds f.x = r)
holds Integral(M,f) = r * M.(dom f);
theorem :: MEASUR10:28
for X being non empty set,
S being SigmaField of X, M being sigma_Measure of S,
f being PartFunc of X,ExtREAL, A being Element of S
st (ex E being Element of S st E = dom f & f is_measurable_on E) &
(for x being object st x in dom f \ A holds f.x = 0) &
f is nonnegative
holds Integral(M,f) = Integral(M,f|A);
theorem :: MEASUR10:29
for X being non empty set,
S being SigmaField of X, M being sigma_Measure of S,
f being PartFunc of X,ExtREAL, A being Element of S
st f is_integrable_on M &
(for x being object st x in dom f \ A holds f.x = 0)
holds Integral(M,f) = Integral(M,f|A);
theorem :: MEASUR10:30
for X1,X2 being non empty set,
S1 being SigmaField of X1,
S2 being SigmaField of X2,
M2 being sigma_Measure of S2,
D being Function of NAT,S1, E being Function of NAT,S2,
A being Element of S1, B being Element of S2,
F being Functional_Sequence of X2,ExtREAL,
RD being sequence of Funcs(X1,REAL),
x being Element of X1 st
(for n being Nat holds RD.n = chi(D.n,X1))
& (for n being Nat holds F.n = ((RD.n).x)(#)(chi(E.n,X2)))
& (for n being Nat holds E.n c= B)
holds
ex I be ExtREAL_sequence st
(for n be Nat holds I.n = M2.(E.n) * (chi(D.n,X1).x)) &
I is summable & Integral(M2,lim Partial_Sums F) = Sum I;
theorem :: MEASUR10:31
for X being non empty set, S being SigmaField of X, A being Element of S,
p being R_eal holds
X --> p is_measurable_on A;
definition
let A,X be set;
func Xchi(A,X) -> Function of X,ExtREAL means
:: MEASUR10:def 7
for x being object st x in X holds
(x in A implies it.x = +infty) & (not x in A implies it.x = 0);
end;
theorem :: MEASUR10:32
for X being non empty set, S being SigmaField of X, A,B being Element of S
holds Xchi(A,X) is_measurable_on B;
registration
let X,A be set;
cluster Xchi(A,X) -> nonnegative;
end;
theorem :: MEASUR10:33
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
A be Element of S holds
(M.A <> 0 implies Integral(M,Xchi(A,X)) = +infty) &
(M.A = 0 implies Integral(M,Xchi(A,X)) = 0);
theorem :: MEASUR10:34
for X1,X2 being non empty set, S1 being SigmaField of X1,
S2 being SigmaField of X2,
M1 being sigma_Measure of S1, M2 being sigma_Measure of S2,
K being disjoint_valued Function of NAT,measurable_rectangles(S1,S2) st
Union K in measurable_rectangles(S1,S2) holds
product-pre-Measure(M1,M2).(Union K) = SUM(product-pre-Measure(M1,M2)*K);
theorem :: MEASUR10:35
for f be without-infty FinSequence of ExtREAL,
s be without-infty ExtREAL_sequence
st (for n be object st n in dom f holds f.n = s.n) holds
Sum f + s.0 = (Partial_Sums s).(len f);
theorem :: MEASUR10:36
for f be nonnegative FinSequence of ExtREAL, s be ExtREAL_sequence
st (for n be object st n in dom f holds f.n = s.n) &
(for n be Element of NAT st not n in dom f holds s.n = 0) holds
Sum f = Sum s & Sum f = SUM s;
theorem :: MEASUR10:37
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
M1 be sigma_Measure of S1, M2 be sigma_Measure of S2 holds
for F be disjoint_valued FinSequence of measurable_rectangles(S1,S2)
st Union F in measurable_rectangles(S1,S2) holds
product-pre-Measure(M1,M2).(Union F)
= Sum(product-pre-Measure(M1,M2)*F);
theorem :: MEASUR10:38
for X1,X2 be non empty set,
S1 be SigmaField of X1, S2 be SigmaField of X2,
M1 be sigma_Measure of S1, M2 be sigma_Measure of S2 holds
product-pre-Measure(M1,M2) is pre-Measure of measurable_rectangles(S1,S2);
definition
let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
M1 be sigma_Measure of S1, M2 be sigma_Measure of S2;
redefine func product-pre-Measure(M1,M2) -> pre-Measure of
measurable_rectangles(S1,S2);
end;