In this paper we define the consistent criteria of hypotheses such as the probability of any kind of errors is zero for given criteria. We prove necessary and sufficient conditions for the existence of such criteria.

Consistent criteriasingularorthogonalweakly separablestrongly separable probability measures62H0562H12Basic notions and consistent criterion on hypothesis for countable family of probability measures

Let (E,S) be a measurable space with a given family of probability measures: {μi,i∈I}.

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:

∀i∈I∀j∈Iμi(Xj)=1ifi=j,0ifi≠j.

∀i∈I∀j∈Icard(Xi∩Xj)<c if i≠j, where c denotes a power of continuum.

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∈Iμi(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.

Let H be set of hypotheses and B(H) be σ-algebra of subsets of H which contains all finite subsets of H.

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. □

LetH={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. □

LetH={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. □

LetH={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. □

It follows from the Theorems 3, 4, that for the countable family of probability measures {μHi,i∈N}the notions of weakly separable, separable, orthogonal and strong separable are equivalent.

Consistent criteria in Banach space

Let Mσ be a real linear space of all alternating finite measures on S.

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

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;

if μ∈MB, |f(x)|≤1, than νf(A)=∫Af(x)μ(dx)∈MB, and ‖νf‖≤‖ν‖, where f(x) is a real measurable function, A∈S;

if νn∈MB, νn>00$]]>, νn(E)<∞, n=1,2,… and νn↓0, then for any linear functional l∗∈MB∗limn→∞l∗(νn)=0.

The construction of the Banach space of measures is studied in paper [8]. The following theorem have also been proved in this paper:

LetMBbe a Banach space of measures, then inMBthere exists a family of pairwise orthogonal probability measures{μi,i∈I}such thatMB=⨁i∈IMB(μi),whereMB(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 {Hi} be a countable family of hypotheses. Denote by F=F(MB) the set of real functions f for which ∫Ef(x)μHi(dx) is defined for all μHi∈MB, where MB=⨁i∈NMB(μHi).

LetMB=⨁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 correspondencef→lfdefined by the equality∫Ef(x)μH(dx)=lf(μH),∀μH∈MB,is one-to-one.

Herelfis the linear functional onMB,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>00$]]> a.e. with respect to the measure μHi and
∫EfHi(x)μHi(dx)<∞,μHi(C)=∫CfHi(x)μHi(dx),
then
∫EfHi(x)μHj(dx)=lfHi(μHj)=0,∀j∈N.
Denote CHi={x∣fHi(x)>0}0\}$]]>, 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.

In what follows B(E,S) will always denote a vector space formed by all real bounded measurable functions on (E,S) having the natural order. It is an (AN)-space with identity according to which a function is identically equal to one on E (see [5]). Let B′(E,S) denote the topological conjugate space of B(E,S), which is an order-complete Banach lattice. The elements of B′(E,S) are called finitely-additive measures on (E,S) and the canonical bilinear form which puts B(E,S) and B′(E,S) in duality is denoted by
⟨f,μ⟩=μ(f)=∫Ef(x)μ(dx),f∈B(E,S),μ∈B′(E,S)
and called the integral of f with respect to μ. In what follows B(H,B(H)) is the space of measurable bounded functions and B′(H,B(H)) is the conjugate space of all finitely-additive measures on (H,B(H)).

Equal units of B(H,B(H)) space are denoted by eH and elements B(H,B(H)) and B′(H,B(H)) are denoted by g and ν respectively. Intersection of {ν∈B′(H,B(H))∣⟨eH,ν⟩=1} and positive cone is denoted by SH. It is clear that SH is a compact subset of the simple share, so a set of extreme points of this cone is not empty.

It is also well known that in the (ZFC), (CH), (MA) theory there exists a continual weekly separable family of probability measures which is not strongly separable. Here and in the sequel we denote by (MA) the Martin’s axiom (see [3]).

LetMB=⨁H∈HMB(μH)be the Banach space of measures, E be the complete separable metric space, S be the Borel σ-algebra in E andcardH≤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 correspondencef→lfby the equality∫Ef(x)μH(dx)=lf(μH),∀μH∈MBis one-to-one. Herelfis a linear continuous functional onMB,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:

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

BH⊂XH for all H<ωα.

BH∩BH′=∅ for all H<ωα, H′<ωα, H≠H′.

μH(BH)=1 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. □

ReferencesBorovkov, A.A.: Ibramkhalilov, Sh. Skorokhod A. V, I.: Jech, T.: Kharazishvili, A.B.: Maleev, M., Serechenko, A., Tarashchanskii, M.: Family of probability measures admitting consistent estimators. In: Skorokhod, A.V. (ed.) Zerakidze, Z.: On weakly divisible and divisible families of probability measures. Zerakidze, Z.: On consistent estimators for families of probability measures. In: Zerakidze, Z.S.: Banach space of measures. In: Grigelionis, B., et al. (eds.)