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Induced measures on µ** -measurable setsPeter S. Chami a & Norris Sookoo ba Department of Computer Science, Mathematics and Physics , Faculty of AppliedSciences The University of the West Indies , Cavehill, St. Michael , Barbadosb The University of Trinidad and Tobago O’Meara Campus , Lots # 74–98 O’MearaIndustrial Park Arima, Trinidad , West IndiesPublished online: 28 May 2013.

To cite this article: Peter S. Chami & Norris Sookoo (2010) Induced measures on µ** -measurable sets, Journal ofInterdisciplinary Mathematics, 13:6, 691-702, DOI: 10.1080/09720502.2010.10700727

To link to this article: http://dx.doi.org/10.1080/09720502.2010.10700727

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Induced measures on µ∗∗ -measurable sets

Peter S. Chami ∗2

Department of Computer Science, Mathematics and PhysicsFaculty of Applied Sciences4

The University of the West IndiesCavehill, St. Michael6

Barbados

Norris Sookoo8

The University of Trinidad and TobagoO’Meara Campus10

Lots # 74-98 O’Meara Industrial ParkArima, Trinidad12

West Indies

Abstract14

We investigate extension of a measure to a very general set of undetermined structure.Structure may be imposed on this set in special cases.16

Keywords and phrases : Lebesgue, topological spaces, Lukasiewicz tribe, σ -algebra, measures.

1. Introduction18

Interest in the extension has continued into recent times with atten-tion being paid to special types of measures, or to extension into particular20

types of sets, rather than just the general theory of extension of measures.[2] showed that the Lebesgue-like extension of every finitely measure on22

the Cartesian product of a countable number of discrete topological spacesare a measure on the lattice of open sets. [6] Considered G -invariant24

measures, where G is and at most countable group of bijections. [3]Established that each sequentially continuous normed measure on a bold26

∗E-mail: peter.chami@cavehill.uwi.edu

——————————–Journal of Interdisciplinary MathematicsVol. 13 (2010), No. 6, pp. 691–702c© Taru Publications

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692 P. S. CHAMI AND N. SOOKOO

algebra of fuzzy sets can be uniquely extended to sequentially continuousmeasure on the generated Lukasiewicz tribe. [4] Also characterized ex-2

tension of probability measures as a completely categorical construction.We consider extension of a measure onto a very general set, so there is4

great variety in its possible structure. In a particular situation, appropriatestructure can be imposed.6

A measure µ on a ring R induces an outer measure µ∗ on thesmallest σ -set H(R) containing R . µ∗ in turn induces a complete8

measure on S̄ , the σ -algebra of all µ∗ -measurable sets (c.f. [5]). Weinvestigate whether µ may induce a measure on a very general set. We10

show that µ induces a complete measure on a set containing S̄ whicharises in a natural way by considering sets whose intersections with12

µ∗ -measurable sets are also µ∗ -measurable. This was suggested by thedefinition of sets measurable with respect to a measurable space (c.f. [1]).14

The results obtained are generalizations of results from [5].

2. Definitions and notations16

Definition. Let (G,B) be a measurable space. Let A ⊂ G , and supposethat for every B ∈ B , we have A ∩ B ∈ B . We then say that A18

is measurable with respect to (G, B) . We denote the totality of suchmeasurable sets A by B̃ . (c.f. [1]).20

The following definition and two notations are from [5].

Notation. For any class E of sets, S(E) is the smallest σ -ring contain-22

ing E .

Definition. A non-empty class E of sets is hereditary if24

F ⊂ E, whenever E ⊂ E and F ⊂ E.

Notation. H(R) denotes the smallest hereditary set containing a ring R .26

Notation. Let X be a set having non-empty intersection with the setcontaining all elements occurring in the sets of R .28

Notation. Let K = {B ∈ X|B ∩ E ∈ H(R) for all E ∈ H(R)} .

Definition. A set Q ∈ K is µ∗∗ -measurable if Q ∩ E is µ∗ -measurable30

∀ E ∈ H 3 E is µ∗ -measurable; that is , if ∀ A ∈ H and for anyµ∗ -measurable set E ,32

µ∗(A) = µ∗[A ∩ (Q ∩ E)] +µ∗[A ∩ (Q ∩ E)c] .

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INDUCED MEASURES ON µ∗∗ -MEASURABLE SETS 693

3. Induced measures

Theorem 3.1. If µ∗ is an outer measure on a hereditary σ -ring H and if ¯̄S is2

the class of all µ∗∗ -measurable sets, then ¯̄S is a ring.

Proof. Let L, M ∈ ¯̄S . Then for any G ∈ S̄ , L ∩ G ∈ S̄ and M ∩ G ∈ S̄ .4

Therefore

(L ∪ M) ∩ G ∈ S̄ .6

Therefore

L ∪ M ∈ ¯̄S . (I)8

We now show that (M − L) ∩ G ∈ S̄ , ∀ G ∈ S̄ .Since M ∩ G is µ∗ -measurable,10

µ∗(A) = µ∗[A ∩ (M ∩ G)] +µ∗[A ∩ (M ∩ G)′] . (II)

Since L ∩ G is µ∗ -measurable,12

µ∗[A ∩ (M ∩ G)] = µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)′] . (III)14

Also,

µ∗[A ∩ (M ∩ G)′] = µ∗[(A ∩ (M ∩ G)′ ∩ (L ∩ G)]16

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)′] . (IV)

Substituting (III) and (IV) into (II)18

µ∗(A) = µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)′]20

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)′] . (V)22

It is not difficult to establish that:

A ∩ (M ∩ G) ∩ (L ∩ G)′ = A ∩ [(M − L) ∩ G] . (VI)24

Also, substituting A ∩ [(M ∩ L′) ∩ G]′ in place of A in (V).

µ∗{A ∩ [(M − L) ∩ G]′}26

= µ∗{A ∩ [(M ∩ L′) ∩ G]′}= µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)}28

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694 P. S. CHAMI AND N. SOOKOO2

+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ [(M ∩ G) ∩ (L ∩ G)′]}+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)}4

+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)′} . (VII)

We can easily prove (1), (2), (3) and (4) below.6

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)

= A ∩ (M ∩ G) ∩ (L ∩ G) , (1)8

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)′ = φ , (2)

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)10

= A ∩ (M ∩ G)′ ∩ (L ∩ G) , (3)

A ∩ [(M ∪ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)′12

= A ∩ (M ∩ G)′ ∩ (L ∩ G)′ . (4)

From (V), (VI), (1) ,(2), (3), and (4)14

µ∗(A) = µ∗{An[(M − L)nG]}+µ∗{A ∩ [(M − L) ∩ G]′} .

Hence (M − L) ∩ G is µ∗ -measurable for any G ∈ S̄ .16

Hence (M − L) is µ∗∗ -measurable, i.e.,

(M − L) ∈ ¯̄S . (VIII)18

From (I) and (VIII) ¯̄S is a ring. ¤

Theorem 3.2. If µ∗ is an outer measure on a hereditary σ -ring H and if ¯̄S is20

in the class of all µ∗∗ -measurable sets, then ¯̄S is a σ -ring.

Proof. Let L1, L2, . . . be an infinite sequence of sets in ¯̄S Then for any22

G ∈ S̄, Li ∩ G ∈ S̄; i = 1, 2, . . . ,

(L1 ∪ L2 ∪ . . .) ∩ G = (L1 ∩ G) ∪ (L2 ∩ G) ∪ . . . ∈ S̄ ,24

since S̄ is an r -ring (Theorem B).

Hence (L1 ∪ L2 ∪ . . .) ∈ ¯̄S .26

Hence ¯̄S is a σ -ring. ¤

Definition. Let µ̃ be a set function defined on ¯̄S by28

µ̃(P) = supT∈S̄

µ∗(P ∩ T), ∀ P ∈ ¯̄S .

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INDUCED MEASURES ON µ∗∗ -MEASURABLE SETS 695

Lemma 3.3. If A ∈ H and if {Ln} is a disjoint sequence of sets in ¯̄S with∞⋃

n=1Ln = L , then2

supT∈S̄

∞∑

n=1µ∗(A ∩ Ln ∩ T) =

∞∑

n=1supT∈S̄

µ∗(A ∩ Ln ∩ T) .

Proof. Let µ∗(A ∩ Tn ∩ T) take its maximum value for T = Tn and let4

∞⋃n=1

Tn = U . Then U ∈ S̄ and

supT∈S̄

µ∗(A ∩ Ln ∩ T) = µ∗(A ∩ Ln ∩ U) . (I)6

Suppose that ∃ V ∈ S̄ 3 ∞∑

n=1µ∗(A ∩ Ln ∩ V) >

∞∑

n=1µ∗(A ∩ Ln ∩ U) .

Then ∃ some value of n , N says, 3 µ∗(A ∩ Ln ∩ V) > µ∗(A ∩ Ln ∩ U) ,8

contradicting (I).

Hence10

supT∈S̄

∞∑

n=1µ∗(A ∩ Ln ∩ T) =

∞∑

n=1µ∗(A ∩ Ln ∩ U)

=∞∑

n=1supT∈S̄

µ∗(A ∩ Ln ∩ T) . ¤12

Theorem 3.4. If A ∈ H and if {Ln} is a disjoint sequence of sets in ¯̄S with∞⋃

n=1Ln = L , then14

µ̃(A ∩ L) =∞∑

n=1µ̃(A ∩ Ln) .

Proof. Let T be an arbitrary element of S̄ . Then {Ln ∩ T} is a disjoint16

sequence of sets in S̄ and L ∩ T =∞V

n=1(Ln ∩ T) .

Hence18

µ∗(A ∩ L ∩ T) =∞∑

n=1µ∗(A ∩ Ln ∩ T)

from Theorem 11.B, by [5], therefore20

supT∈S̄

∞∑

n=1µ∗[A ∩ (Ln) ∩ T] =

∞∑

n=1supT∈S̄

µ∗(A ∩ Ln ∩ T)

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696 P. S. CHAMI AND N. SOOKOO

from the previous lemma.

I.e., µ̃(A ∩ L) =∞∑

n=1µ̃(A ∩ Ln) . ¤2

Definition. The set function µ̄ is defined on S̄ by

µ̄(E) = µ∗(E), for E ∈ S̄.4

Remark. µ̄ is a complete measure on S̄ (Theorem 11(c), [5]).

Lemma 3.5.6

supT∈S̄

{ ∞∑

n=1µ̄(Ln ∩ T)

}=

∞∑

n=1supT∈S̄

µ̄(Ln ∩ T) ,

where {Ln} is a disjoint sequence in ¯̄S .8

Proof. As S̄ is monotone, it is easy to show that ∃ Tn 3 µ̄(Ln ∩ T) takes

its maximum value for T = Tn . Let U =∞⋃

n=1Tn . Then U ∈ S̄ and10

supT∈S̄

µ̄(Ln ∩ T) = µ̄(Ln ∩ U)

Therefore12

∞∑

n=1supT∈S̄

µ̄(Ln ∩ T) =∞∑

n=1µ̄(Ln ∩ U)

= µ̄

[( ∞⋃

n=1Ln

)∩ U

]14

= supT∈S̄

µ̄

[( ∞⋃

n=1Ln

)∩ T

]

= supT∈S̄

{ ∞∑

n=1µ̄(Ln ∩ T)

}.16

Theorem 3.6. If µ∗ is an outer measure on a hereditary σ -ring H and if ¯̄S isthe class of all µ∗∗ -measurable sets, then every set of outer measure zero belongs18

to ¯̄S and µ̃ is a complete measure on ¯̄S .

Proof. If E ∈ H and µ∗(E) = 0 , then for any G ∈ S̄ and A ∈ H ,20

µ∗(A) = µ∗(E) +µ∗(A)

≥ µ∗[A ∩ (E ∩ G)] +µ∗[A ∩ (E ∩ G)c]22

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INDUCED MEASURES ON µ∗∗-MEASURABLE SETS 697

Since

µ∗(A) ≤ µ∗[A ∩ (E ∩ G)] +µ∗[A ∩ (E ∩ G)c],2

µ∗(A) = µ∗[A ∩ (E ∩ G)] +µ∗[A ∩ (E ∩ G)c].

Hence E ∈ ¯̄S .4

Countable additivity. Let {Ln} be a disjoint sequence in ¯̄S . For anyT ∈ S̄ ,6

µ̄

[( ∞⋃

n=1Ln

)∩ T

]= µ̄

[ ∞⋃

n=1(Ln ∩ T)

]

=∞∑

n=1µ̄(Ln ∩ T)8

since µ̄ is a complete measure on S̄ . (Theorem 11(c) [5]).

Therefore10

supT∈S̄

µ̄

[( ∞⋃

n=1Ln

)∩ T

]= sup

T∈S̄

{ ∞∑

n=1µ(Ln ∩ T)

}

=∞∑

n=1supT∈S̄

µ̄(Ln ∩ T)12

(by the previous lemma).

Therefore14

supT∈S̄

µ∗[( ∞⋃

n=1Ln

)∩ T

]=

∞∑

n=1supT∈S̄

µ∗(Ln ∩ T)

i.e.,16

µ̃

( ∞⋃

n=1LN

)=

∞∑

n=1µ̃(Ln) .

Therefore µ̃ is count ably additive and hence a measure.18

Completeness. If E ∈ ¯̄S , F ⊂ C and µ̃(E) = 0 , then µ̃(F) = 0 and soF ∈ ¯̄S . Hence µ̃ is complete. ¤20

Remark. µ̃ is called the measure induced by µ∗ .

Theorem 3.7. Every set in S(R) is µ∗∗ -measurable.22

Proof. Let E ∈ S̄ . For any element G of S̄ , E ∩ G ∈ S̄ . Hence E isµ∗∗ -measurable.24

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698 P. S. CHAMI AND N. SOOKOO

Therefore

E ∈ ¯̄S .2

Therefore

S̄ ⊂ ¯̄S .4

Since R ⊂ S̄ (Theorem A, p. 49, [5]), R ⊂ ¯̄S . Since ¯̄S is a σ -ring.

S(R) ⊂ ¯̄S .6

Theorem 3.8. If E ∈ H(R) , then

µ∗(E) = inf{µ̃(F) : E ⊂ F ∈ S̄} ,8

= inf{µ̃(F) : E ⊂ F ∈ S(R)} .

Proof. Recall that10

µ∗(F) = inf{ ∞

∑n=1

µ(En) : En ∈ R and E ⊂∞⋃

n=1En

}.

If F ∈ R , then, by the above definition, µ∗(F) = µ(F) . Since F ∈ R ,12

F ∈ S̄ , F ∈ ¯̄S .

Therefore14

µ̃(F) = supT∈S̄

µ∗(F ∩ T) = µ∗(F) .

Therefore, if F ∈ R ,16

µ(F) = µ̃(F) .

From (I),18

µ∗(E) ≥ inf{ ∞

∑n=1

µ̃(En) : En ∈ S(R), and E ⊂∞⋃

n=1En

}.

Since every sequence {En} of sets in S(R) for which20

E ⊂∞⋃

n=1En = F

may be replaced by a disjoint sequence with the same property without22

increasing the sum of the measures of the terms of the sequence, and since,

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INDUCED MEASURES ON µ∗∗ -MEASURABLE SETS 699

by the definition of µ̃ ,

µ̃(E) ≥ µ∗(F), ∀ F ∈ S̄2

it follows that

µ∗(E) ≥ inf{µ̃(F) : E ⊂ F ∈ S(R)}4

≥ inf{µ̃(F) : E ⊂ F ∈ S̄}≥ inf{µ∗(F) : E ⊂ F ∈ S̄}6

≥ µ∗(E)

and the result follows. ¤8

Remark. Given n measure spaces (Xi , Si ,µi) , with the Si , s beingmutually disjoint, the measure a µT can be defined on the space T, where10

T =

{ n⋃

i=1

Ai | Ai ∈ Si , i = 1, 2, . . . , n}

.

and12

µT

( n⋃

i=1

Ai

)=

n

∑i=1

Ai .

T can be defined in a variety of other ways.14

We can also consider the outer measure µ∗i and set function

µ̃i associated with each µi , i = 1, 2, . . . , n , and investigate the measure16

induced by the µi ’s.

Appendix18

Detailed Proof of Theorem 3.1

Proof. Let L, M ∈ ¯̄S . Then for any G ∈ S̄ , L ∩ G and M ∩ G ∈ S̄ .20

Now, (L ∪ M) ∩ G = (L ∩ G) ∪ (M ∩ G) .Since L ∩ G, M ∩ G ∈ S̄ and S̄ is a ring.22

(L ∩ G) ∪ (M ∩ G) ∈ S̄ .

Therefore24

(L ∪ M) ∩ G ∈ S̄ . (I)

Therefore26

L ∪ M ∈ ¯̄S .

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700 P. S. CHAMI AND N. SOOKOO

We now show that (M − L) ∩ G ∈ S̄ , ∀ G ∈ S̄ .Since M ∩ G is µ∗ -measurable,2

µ∗(A) = µ∗[A ∩ (M ∩ G)] +µ∗[A ∩ (M ∩ G)′] . (II)

Since L ∩ G is µ∗ -measurable,4

µ∗[A ∩ (M ∩ G)] = µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)′] . (III)6

Also,

µ∗[A ∩ (M ∩ G)′] = µ∗[(A ∩ (M ∩ G)′ ∩ (L ∩ G)]8

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)′] . (IV)

Substituting (III) and (IV) into (II)10

µ∗(A) = µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G) ∩ (L ∩ G)′]12

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)]

+µ∗[A ∩ (M ∩ G)′ ∩ (L ∩ G)′] . (V)14

Now

A ∩ (M ∩ G) ∩ (L ∩ G)′ = A ∩ (M ∩ G) ∩ (L′ ∪ G′)16

= (A ∩ M ∩ G ∩ L′) ∪ (A ∩ M ∩ G ∩ G′

= A ∩ [(M ∩ L′) ∩ G]18

= A ∩ [(M − L) ∩ G] . (VI)

Also,20

µ∗{A ∩ [(M − L) ∩ G]′}= µ∗{A ∩ [(M ∩ L′) ∩ G]′}22

= µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)}+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ [(M ∩ G) ∩ (L ∩ G)′]}24

+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)}+µ∗{A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)′}, (VII)26

substituting A ∩ [(M ∩ L′) ∩ G]′ in place of A in (V).

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)28

= A ∩ [(M ∩ G)′ ∪ L] ∩ (M ∩ G) ∩ (L ∩ G)

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INDUCED MEASURES ON µ∗∗-MEASURABLE SETS 7012

= A ∩ (M ∩ G) ∩ (L ∩ G), (1)

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G) ∩ (L ∩ G)′4

= A ∩ (M′ ∪ L ∪ G′) ∩ (M ∩ G) ∩ (L′ ∪ G′)

= A ∩ (M′ ∪ L ∪ G′) ∩ (M ∩ G ∩ L′)6

= A ∩ (M ∩ G ∩ L′)′ ∩ (M ∩ G ∩ L′)

= φ , (2)8

A ∩ [(M ∩ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)

= A ∩ (M′ ∪ L ∪ G′) ∩ (M′ ∪ G′) ∩ (L ∩ G)10

= A ∩ (M′ ∪ L ∪ G′) ∩ (M′ ∩ L ∩ G)

= A ∩ (M′ ∩ L ∩ G) ∪ (M′ ∩ L ∩ G) ∪φ12

= A ∩ (M′ ∩ L ∩ G′)

= A ∩ (M ∩ G)′ ∩ (L ∩ G), (3)14

A ∩ [(M ∪ L′) ∩ G]′ ∩ (M ∩ G)′ ∩ (L ∩ G)′

= A ∩ (M′ ∪ L ∪ G′) ∩ (M′ ∪ G′) ∩ (L′ ∪ G′)16

= A ∩ (M′ ∪ L ∪ G′) ∩ {[M′ ∩ (L′ ∪ G′)] ∪ [G′ ∩ (L′ ∪ G′)]}= A ∩ [(M ∩ G)′ ∪ L] ∩ {[M′ ∩ (L ∩ G)′] ∪ [G ∩ (L ∩ G)′]}18

= A ∩ [(M ∩ G)′ ∪ L] ∩ [[{M′ ∪ [G′ ∩ (L ∩ G)′]}∩{(L ∩ G)′ ∪ [G′ ∩ (L ∩ G)′]}]]20

= A ∩ [(M ∩ G)′ ∪ L] ∩ [(M ∩ G)′ ∩ (L ∩ G)′]

= A ∩ {[(M ∩ G)′ ∩ (L ∩ G)′] ∪ [L ∩ (M ∩ G)′ ∩ (L ∩ G)′]}22

= A ∩ {[(M ∩ G)′ ∩ (L ∩ G)′] ∪ [L ∩ (M ∪ G)′ ∩ (L′ ∪ G′)]}= A ∩ {[(M ∩ G)′ ∩ (L ∩ G)′] ∪ [(M ∪ G)′ ∩ (L ∩ G′)]}24

= A ∩ {(M ∩ G)′ ∩ [(L ∩ G)′ ∪ (L ∩ G′)]}= A ∩ {(M ∩ G)′ ∩ [(L′ ∪ G′) ∪ (L ∩ G′)]}26

= A ∩ {(M ∩ G)′ ∩ [U ∩ (L′ ∩ G′)]}= A ∩ (M ∩ G)′ ∩ (L ∩ G)′ . (4)28

From (V), (VI), (1) ,(2), (3), and (4)

µ∗(A) = µ∗{An[(M − L)nG]}+µ∗{A ∩ [(M − L) ∩ G]′} .30

Hence (M − L) ∩ G is µ∗ -measurable for any G ∈ S̄.

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702 P. S. CHAMI AND N. SOOKOO

Hence (M − L) is µ∗∗ -measurable, i.e.,

(M − L) ∈ ¯̄S . (VIII)2

From (I) and (VIII) ¯̄S is a ring. ¤

References4

[1] X. Dao-Xing (1972), Measure and Integration Theory on Infinite-Dimensional Spaces, translated by Elmer J. Brody, Academic Press.6

[2] L. E. Dubins (1974), On the Lebesgue-like extension of finitelyadditive measures, Ann. Probab., Vol. 2 (3), pp. 456–463.8

[3] F. Fric (2002), Lukasiewicz tribes are absolutely sequentially closedbold algebras, Czechoslovak Math. J., Vol. 52 (4), pp. 861–874.10

[4] R. Fric (2005), Extension of measures: a categorical approach, Math.Bohem., Vol. 130 (4), pp. 397–407.12

[5] P. R. Halmos (1950), Graduate Texts in Mathematics – Measure Theory,Springer-Verlag.14

[6] A. Krawczyk and P. Zakrzewski (1991), Extensions of measuresinvariant under countable groups of transformations, Trans. Amer.16

Math. Soc., Vol. 326 (1), pp. 211–226.

Received February, 201018

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