The family {μi,i∈I} of probability measures is called orthogonal (singular) if μi and μj are orthogonal for each i≠j .

The family {μi,i∈I} of probability measures is called separable if there exists a family of S-measurable sets {Xi,i∈I} such that the relations are fulfilled:

(1)

∀i∈I ∀j∈I μi(Xj)=1ifi=j,0ifi≠j.

The family {μi,i∈I} of probability measures is called weakly separable if there exists a family of S-measurable sets {Xi,i∈I} such that the relations are fulfilled: ∀i∈I∀j∈Iμi(Xj)=1ifi=j,0ifi≠j.

The family {μi,i∈I} of probability measures is called strongly separable if there exists a disjoint family of S-measurable sets {Xi,i∈I} such that the relations are fulfilled: ∀i∈I μi(Xi)=1 .

A strong separability implies separability, a separability implies a weak separability, and a weak separability implies orthogonality, but not vice versa.

Let E=[0,1]×[0,1] , S be a Borel σ-algebra of subsets of E. Take the S-measurable sets Xi={0≤x≤1,y=i,i∈[0,1]} and assume that Li are the linear Lebesgue probability measures on Xi . Then the family {Li,i∈[0,1]} is strongly separable.

Let E=[0,1]×[0,1] , S be a Borel σ-algebra of subsets of E. Take the S-measurable sets Xi={(x,y)∣0≤x≤1,y=i,ifi∈[0,1];x=i−2,0≤y≤1,ifi∈[2,3]}. Let Li be the linear Lebesgue probability measures on Xi . Then the family {Li,i∈[0,1]∪[2,3]} is separable but not strongly separable.

Let E=[0,1]×[0,1]×[0,1] , S be a Borel σ-algebra of subsets of E. Take the S-measurable sets: Xi={(x,y,z)∣0≤x≤1,0≤y≤1,z=i,ifi∈[0,1];x=i−2,0≤y≤1,0≤z≤1,ifi∈[2,3];0≤x≤1,y=i−4,0≤z≤1,ifi∈[4,5]}. Let Li be the planar Lebesgue probability measures on Xi . Then the family {Li,i∈[0,1]∪[2,3]∪[4,5]} is weakly separable but not separable.

Let E=[0,1]×[0,1] , S be a Borel σ-algebra of subsets of E. Take the S-measurable sets Xi={(x,y)∣0≤x≤1,y=i,i∈(0,1]}. Let Li be the linear Lebesgue probability measures on Xi and L0 be the planar Lebesgue probability measure on E=[0,1]×[0,1] . Then the family {Li,i∈[0,1]} is orthogonal, but not weakly separable.

We consider the notion of Hypothesis as any assumption that defines the form of the distribution selection.

The family of probability measures {μH,H∈H} is said to admit a consistent criteria of a hypothesis if there exists at least one measurable map δ:(E,S)→(H,B(H)) , such that μH({x∣δ(x)=H})=1 , for all H∈H .

The following probability: αH(δ)=μH({x∣δ(x)≠H}) is called the probability of error of the H-th kind for a given criterion δ.

The family of probability measures {μH,H∈H} is said to admit a consistent criterion of any parametric function if for any real bounded measurable function g:(H,B(H))→R there exists at least one measurable function f:(E,S)→R such that μH({x∣f(x)=g(H)})=1 , for all H∈H .

The family of probability measures {μH,H∈H} is said to admit an unbiased criterion of any parametric function if for any real bounded measurable function g:(H,B(H))→R there exists at least one measurable function β:(E,S)→R , such that ∫Eβ(x)μH(dx)=g(H) for all H∈H .

If M is a family of probability measures admitting a consistent criterion for a hypothesis, then it is clear that M is a family of probability measures which admits a consistent criterion for any parametric function and a family of probability measures which admits an unbiased criterion of any parametric function.

The family of probability measures {μH,H∈H} admits a consistent criterion δ of a hypothesis if and only if the probability of error of all kinds is equal to zero for the criterion δ.

Necessity. As the family of probability measures {μH,H∈H} admits a consistent criterion of a hypothesis, so there exists such a measurable map δ:(E,S)→(H,B(H)) that μH({x∣δ(x)=H})=1 for all H∈H . It follows αH(δ)=μH({x∣δ(x)≠H}=0 .

Sufficiency. As the probability of error of all kinds is equal to zero, so αH(δ)=μH({x∣δ(x)≠H}=0 for all H∈H , we have {x∣δ(x)=H}∩{x∣δ(x)=H′}=∅ for any H≠H′ .

On the other hand {x∣δ(x)=H}∪{x∣δ(x)≠H}=E and μH({x∣δ(x)=H})=1 , for all H∈H .

Therefore δ is a consistent criterion of a hypothesis. The Theorem 1 is proved.  □

Let H={H1,H2,…,Hn,…} be the set of hypotheses. The family of probability measures {μHi,i∈N} , N={1,2,…,n,…} admits the consistent criterion of hypotheses if and only if the family of probability measures {μHi,i∈N} is strongly separable.

Necessity. Since the family {μHi,i∈N} admits a consistent criterion of hypotheses, then there exists a measurable map δ of the space (E,S) to (H,B(H)) such that μHi(x∣δ(x)=Hi)=1 , i∈N . Let Xi=(x:δ(x)=Hi) , then it is obvious, that Xi∩Xi≠∅ for all i≠j and μHi(Xi)=1 , ∀i∈N . Therefore, the family of probability measures {μHi,i∈N} is strongly separable.

Sufficiency. As the family of probability measures {μHi,i∈N} is strongly separable, then there exist such pairwise disjoint S-measurable sets Xi , i∈N that μHi(Xi)=1 , ∀i∈N .

Let’s define δ as such a mapping (E,S)→(H,B(H)) that δ(Xi)=Hi , i∈N . We have {x:δ(x)=Hi}=Xi and μHi{x:δ(x)=Hi}=1 , ∀i∈N . Therefore δ is a consistent criterion of hypotheses. The Theorem 2 is proved.  □

Let H={H1,H2,…,Hn,…} and the family of probability measures {μHi,i∈N} be separable or weakly separable. Then the family of probability measures {μHi,i∈N} admits a consistent criterion of hypotheses.

Since the family of probability measures {μHi,i∈N} is separable or weakly separable, then there exists a family X1,X2,…,Xn,… of S-measures sets such that μHi(Xj)=1,ifi=j,0,ifi≠j.

Let us consider the sets: X‾1=X1−X1∩(⋃k≠1Xk)X‾2=X2−X2∩(⋃k≠2Xk)…X‾n=Xn−Xn∩(⋃k≠nXk)… It is obvious that {X‾1,X‾2,…,X‾n,…} is a disjoint family of S-measurable sets and μHi(X‾i)=1 , ∀i∈N . Therefore, the family of probability measures {μHi,i∈N} is strongly separable and {μHi,i∈N} admits a consistent criterion of hypotheses by the Theorem 1. The Theorem 3 is proved.  □

Let H={H1,H2,…,Hn,…} and the family of probability measures {μHi,i∈N} N={1,2,…,n,…} be orthogonal (singular). Then the family of probability measures {μHi,i∈N} admits a consistent criterion of hypotheses.

The singularity of probability measures implies an existence of the family {Xik} of S-measurable sets such that for any i≠k we have μHk(Xik)=0 and μHi(E−Xik)=0 .

Let us consider the sets Xi=⋃k≠i(E−Xik) , then μHi(Xi)=μHi(⋃k≠i(E−Xik))≤∑k≠iμHi(E−Xik)=0. Therefore, μHi(Xi)=0 ; μHi(E−Xi)=1 . On the other hand, for k≠i we have μHk(E−Xi)=0 . This means that the family of probability measures {μHi,i∈N} is weakly separable. By the Theorem 3 this family of probability measures admits a consistent criterion of hypotheses. The Theorem 4 is proved.  □

A linear subset MB⊂Mσ is called a Banach space of measures if: (1)

a norm can be defined on MB so that MB will be a Banach space with respect to this norm, and for any orthogonal measures μ,ν∈MB and any real number λ≠0 the inequality ‖μ+λν‖≥‖μ‖ is fulfilled;

Let MB be a Banach space of measures, then in MB there exists a family of pairwise orthogonal probability measures {μi,i∈I} such that MB=⨁i∈IMB(μi), where MB(μi) is the Banach space of elements ν of the form: ν(B)=∫Bf(x)μi(dx),B∈S,∫E|f(x)|μi(dx)<∞, with the norm ‖ν‖MB(μi)=∫E|f(x)|μi(dx).

Let MB=⨁i∈NMB(μHi) be a Banach space of measures. The family of probability measures {μHi,i∈N} admits a consistent criteria of hypotheses if and only if the correspondence f→lf defined by the equality ∫Ef(x)μH(dx)=lf(μH),∀μH∈MB, is one-to-one.

Here lf is the linear functional on MB , f∈F(MB) .

Sufficiency. For f∈F(MB) we define the linear continuous functional lf by the equality ∫Ef(x)μHdx)=lf(μH) . Denote as If a countable subset in N , for which ∫Ef(x)μHi(dx)=0 for i∉If . Let us consider the functional lfHi on MB(μHi) to which it corresponds. Then for μH1,μH2∈MB(μHi) we have: ∫EfH1(x)μH(dx)=lfH1(μH2)=∫Ef1(x)f2(x)μHi(dx)=∫fH1(x)μHi(dx). Therefore fH1=f1 a.e. with respect to the measure μHi . Let fHi>0 0$]]> a.e. with respect to the measure μHi and ∫EfHi(x)μHi(dx)<∞,μHi(C)=∫CfHi(x)μHi(dx), then ∫EfHi(x)μHj(dx)=lfHi(μHj)=0,∀j∈N. Denote CHi={x∣fHi(x)>0} 0\}$]]>, then ∫EfHi(x)μHj(dx)=lfHi(μHj)=0 , ∀j∈N . Hence it follows, that μHj(CHi)=0 , ∀j∈N . On the other hand μHi(E−CHi)=0 . Therefore the family {μHi,i∈N} is weekly separable and μHi(CHj)=1,ifi=j,0,ifi≠j.

Let us consider the sets C‾Hi=CHi\CHi∩⋃k≠iCHk . It is obvious that {C‾Hi,i∈N} is a disjunctive family of S-measurable sets and μHi(CHi)=1 , ∀i∈N . Let us define a mapping δ:(E,S)→(H,B(H)) like that δ(C‾Hi)=Hi , ∀i∈N . We have μHi({x∣δ(x)=Hi})=1 , ∀i∈N . Therefore δ is a consistent criterion of hypotheses.

Necessity. Since the family of probability measures {μHi,i∈N} admits a consistent criterion of hypotheses and this family is strongly separable, so there exist S-measurable sets Xi , i∈N , such that μHi(Xj)=1,ifi=j,0,ifi≠j. We put the linear continuous functional lXi into the correspondence to a function IXi∈F(Mβ) by the formula: ∫EIXi(x)μHi(dx)=lIXi(μHi)=‖μHi‖Mβ(μHi). We put the linear continuous functional lfH1 into the correspondence to the function fH1(x)=f1(x)IXi(x)∈F(Mβ). Then for any μH2∈MB(μi) ∫EfH1(x)μH2(dx)=∫Ef1(x)IXi(x)μH2(dx)=∫Ef(x)f1(x)IXi(x)μHi(dx)=lfH1(μH2)=‖μH2‖MB(μHi).

Let Σ={lf} be the collection of extensions of the functional lfH1:Mβ(μH1)→R satisfying the condition lf≤p(x) on those subspaces where they are defined.

Let us introduce a partial ordering on Σ having assumed lf1<lf2 if lf2 is defined on a larger set than lf1 and lf1(μ)=lf2(μ) where both of them are defined.

Let {lfi}i∈I be a linear ordered subset in Σ. Let MB(μHi) be the subspace on which lfHi is defined. Define lf on ⨁i∈IMB(μHi) , having assumed lf(μ)=lfHi(μ) if μ∈MB(μHi) .

It is obvious, that lfHi<lf . Since any linearly ordered subset in Σ has an upper bound, by virtue of Zorn lemma Σ contains a maximal element λ defined on some set X′ satisfying the condition λ(x)≤p(x) for x∈X′ . But X′ must coincide with the entire space MB because otherwise we could extend λ to a wider space by adding, as above, one more dimension.

This contradicts with the maximality of λ and hence X′=MB . Therefore the extension of the functional is defined everywhere.

If we put the linear continuous functional lf into the correspondence to the function f(x)=∑i∈NgHi(x)IXi(x)∈F(MB), then we obtain ∫Ef(x)μH(dx)=‖μH‖=∑i∈N‖μHi‖MB(μHi), where μH=∑i∈N∫EgHiμHi(dx) . The Theorem 6 is proved.  □

It follows from the proven theorem that the indicated above correspondence puts some functions f∈F(MB) into the correspondence to each linear continuous functional lf . If in F(Mβ) we identify functions coinciding with respect to the measures {μHi,i∈N} , then the correspondence will be bijective.

Let MB=⨁H∈HMB(μH) be the Banach space of measures, E be the complete separable metric space, S be the Borel σ-algebra in E and cardH≤c . Then in the theory (ZFC) and (MA) the family of probability measures {μH,H∈H} admits a consistent criteria of hypotheses if and only if the family of probability measures {μH,H∈H} admits an unbiased criterion of any parametric function and the correspondence f→lf by the equality ∫Ef(x)μH(dx)=lf(μH),∀μH∈MB is one-to-one. Here lf is a linear continuous functional on MB , f∈F(MB) .

Necessity. As the family of probability measures {μH,H∈H} admits a consistent criterion of hypotheses, so the family {μH,H∈H} admits an unbiased criterion of any parametric function and it is strongly separable. So, the family {μH,H∈H} is weekly separable. The necessity is proved in the same manner as the necessity of the Theorem 6.

Sufficiency. According to the Theorem 6 a Borel orthogonal family of probability measures {μH,H∈H} , cardH≤c is weakly separable. We represent {μH,H∈H} as an inductive sequence μH<ωα , where ωα denotes the first ordinal number of the power of the set H. Since the family {μH,H∈H} is weakly separable, there exists a family of measurable parts {XH}H<ωα of the space E, such that the following relations are fulfilled: μH(XH′)=1,ifH=H′,0,ifH≠H′ for all H∈[0,ωα) and H′∈[0,ωα) .

We define ωα -sequence of parts BH of the space E so that the following relations are fulfilled: (1)

BH is a Borel subset in E for all H<ωα .

Assume that B0=X0 . Let further the partial sequence {BH′}H′<H be already defined for H<ωα .

It is clear, that μ∗(⋃H′<HBH′)=0 . Thus there exists a Borel subset YH of the space E, such that the following relations are valid: ⋃H′<HBH′⊂YH and μ(YH)=0 . Assume BH=XH∖YH , thereby the ωα sequence of {BH}H<ωα disjunctive measurable subsets of the space E is constructed. Therefore μH(BH)=1 for all H<ωα . As a family of probability measures {μH,H∈H} admits an unbiased criterion for any parametric function, so there exists a subspace G⊂B(E,S) , containing eE unit and B(E,S) can be imagine as a topological sum of G and H0=μ−1(0) , where the functional μ(f)=∫Ef(x)μ(dx),f∈B(E,S),μ∈B′(E,S) and a family {μH,H∈H} is strongly separable, subspace G is a grid towards canonical order on G (see [5]). We assume that S0 is a minimal σ-algebra of subalgebra S, all function on G are measurable towards S0 . Then G⊂B(E,S0)⊂B(E,S) .

Since a subspace G contains eE and represents a grid, then G⊃B(E,S) and that’s why G=B(E,S) .

As family {μH,H∈H} represents a dense subspace of exSH ( exSH are extreme points of SH ), so Iμ is an ideal in the set S0 which contains zero measured sets for all μ∈{μH,H∈H} and consists only of an empty set.

Hence there exist such sets {AH,H∈H} that μH(AH)=1 and AH∩AH′=∅ for a H≠H0 and E=⋃H∈HAH is a set S0 . It follows from the condition of this theorem that for every T∈B(H) in G there exists fT function, which is a consistent criterion of gT parametric function. If A={x∣ft(x)≠0} , then ⋃H∈TAH⊂A , A∩AH=∅ for all H∉T and hence ⋃H∈TAH=A implying that ⋃H∈TAH⊂S0 .

Then, the mapping δ(x)=H if x∈AH for all H∈H is a consistent criterion of hypotheses. The theorem is proved.  □