Modern Stochastics: Theory and Applications

The family $\{\mu _{i},\hspace{2.5pt}i\in I\}$ of probability measures is called orthogonal (singular) if $\mu _{i}$ and $\mu _{j}$ are orthogonal for each $i\ne j$.

The family $\{\mu _{i},\hspace{2.5pt}i\in I\}$ of probability measures is called separable if there exists a family of S-measurable sets $\{X_{i},\hspace{2.5pt}i\in I\}$ such that the relations are fulfilled:

The family $\{\mu _{i},\hspace{2.5pt}i\in I\}$ of probability measures is called weakly separable if there exists a family of S-measurable sets $\{X_{i},\hspace{2.5pt}i\in I\}$ such that the relations are fulfilled:

The family $\{\mu _{i},\hspace{2.5pt}i\in I\}$ of probability measures is called strongly separable if there exists a disjoint family of S-measurable sets $\{X_{i},\hspace{2.5pt}i\in I\}$ such that the relations are fulfilled: $\forall i\in I$ $\mu _{i}(X_{i})=1$.

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

Let $E=[0,1]\times [0,1]$, S be a Borel σ-algebra of subsets of E. Take the S-measurable sets $X_{i}=\{0\le x\le 1,\hspace{0.2778em}y=i,\hspace{0.2778em}i\in [0,1]\}$ and assume that $L_{i}$ are the linear Lebesgue probability measures on $X_{i}$. Then the family $\{L_{i},\hspace{2.5pt}i\in [0,1]\}$ is strongly separable.

Let $E=[0,1]\times [0,1]$, S be a Borel σ-algebra of subsets of E. Take the S-measurable sets Let $L_{i}$ be the linear Lebesgue probability measures on $X_{i}$. Then the family $\{L_{i},\hspace{2.5pt}i\in [0,1]\cup [2,3]\}$ is separable but not strongly separable.

Let $E=[0,1]\times [0,1]\times [0,1]$, S be a Borel σ-algebra of subsets of E. Take the S-measurable sets: Let $L_{i}$ be the planar Lebesgue probability measures on $X_{i}$. Then the family $\{L_{i},\hspace{2.5pt}i\in [0,1]\cup [2,3]\cup [4,5]\}$ is weakly separable but not separable.

Let $E=[0,1]\times [0,1]$, S be a Borel σ-algebra of subsets of E. Take the S-measurable sets Let $L_{i}$ be the linear Lebesgue probability measures on $X_{i}$ and $L_{0}$ be the planar Lebesgue probability measure on $E=[0,1]\times [0,1]$. Then the family $\{L_{i},\hspace{2.5pt}i\in [0,1]\}$ 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 $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ is said to admit a consistent criteria of a hypothesis if there exists at least one measurable map $\delta :(E,S)\to (\mathrm{H},\mathrm{B}(\mathrm{H}))$, such that $\mu _{H}(\{x\mid \delta (x)=H\})=1$, for all $H\in \mathrm{H}$.

The following probability: is called the probability of error of the H-th kind for a given criterion δ.

The family of probability measures $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ is said to admit a consistent criterion of any parametric function if for any real bounded measurable function $g:(\mathrm{H},\mathrm{B}(\mathrm{H}))\to \mathbb{R}$ there exists at least one measurable function $f:(E,S)\to \mathbb{R}$ such that $\mu _{H}(\{x\mid f(x)=g(H)\})=1$, for all $H\in \mathrm{H}$.

The family of probability measures $\{\mu _{H},\hspace{2.5pt}H\in \mathrm{H}\}$is said to admit an unbiased criterion of any parametric function if for any real bounded measurable function $g:(\mathrm{H},\mathrm{B}(\mathrm{H}))\to \mathbb{R}$ there exists at least one measurable function $\beta :(E,S)\to \mathbb{R}$, such that $\int _{E}\beta (x)\mu _{H}(dx)=g(H)$ for all $H\in \mathrm{H}$.

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 $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ 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 $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ admits a consistent criterion of a hypothesis, so there exists such a measurable map $\delta :(E,S)\to (\mathrm{H},\mathrm{B}(\mathrm{H}))$ that $\mu _{H}(\{x\mid \delta (x)=H\})=1$ for all $H\in \mathrm{H}$. It follows $\alpha _{H}(\delta )=\mu _{H}(\{x\mid \delta (x)\ne H\}=0$.

Sufficiency. As the probability of error of all kinds is equal to zero, so $\alpha _{H}(\delta )=\mu _{H}(\{x\mid \delta (x)\ne H\}=0$ for all $H\in \mathrm{H}$, we have for any $H\ne {H^{\prime }}$.

On the other hand $\{x\mid \delta (x)=H\}\cup \{x\mid \delta (x)\ne H\}=E$ and $\mu _{H}(\{x\mid \delta (x)=H\})=1$, for all $H\in \mathrm{H}$.

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

Let $\mathrm{H}=\{H_{1},H_{2},\dots ,H_{n},\dots \}$ be the set of hypotheses. The family of probability measures $\{\mu _{H_{i}},\hspace{2.5pt}i\in \mathbb{N}\}$, $\mathbb{N}=\{1,2,\dots ,n,\dots \}$ admits the consistent criterion of hypotheses if and only if the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ is strongly separable.

Necessity. Since the family $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ admits a consistent criterion of hypotheses, then there exists a measurable map δ of the space $(E,S)$ to $(\mathrm{H},\mathrm{B}(\mathrm{H}))$ such that $\mu _{H_{i}}(x\mid \delta (x)=H_{i})=1$, $i\in \mathbb{N}$. Let $X_{i}=(x:\delta (x)=H_{i})$, then it is obvious, that $X_{i}\cap X_{i}\ne \varnothing $ for all $i\ne j$ and $\mu _{H_{i}}(X_{i})=1$, $\forall i\in \mathbb{N}$. Therefore, the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in N\}$ is strongly separable.

Sufficiency. As the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ is strongly separable, then there exist such pairwise disjoint S-measurable sets $X_{i}$, $i\in \mathbb{N}$ that $\mu _{H_{i}}(X_{i})=1$, $\forall i\in \mathbb{N}$.

Let’s define δ as such a mapping $(E,S)\to (\mathrm{H},\mathrm{B}(\mathrm{H}))$ that $\delta (X_{i})=H_{i}$, $i\in \mathbb{N}$. We have $\{x:\delta (x)=H_{i}\}=X_{i}$ and $\mu _{H_{i}}\{x:\delta (x)=H_{i}\}=1$, $\forall i\in \mathbb{N}$. Therefore δ is a consistent criterion of hypotheses. The Theorem 2 is proved.  □

Let $\mathrm{H}=\{H_{1},H_{2},\dots ,H_{n},\dots \}$ and the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ be separable or weakly separable. Then the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ admits a consistent criterion of hypotheses.

Since the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ is separable or weakly separable, then there exists a family $X_{1},X_{2},\dots ,X_{n},\dots $ of S-measures sets such that

Let us consider the sets: It is obvious that $\{\overline{X}_{1},\overline{X}_{2},\dots ,\overline{X}_{n},\dots \}$ is a disjoint family of S-measurable sets and $\mu _{H_{i}}(\overline{X}_{i})=1$, $\forall i\in \mathbb{N}$. Therefore, the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ is strongly separable and $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ admits a consistent criterion of hypotheses by the Theorem 1. The Theorem 3 is proved.  □

Let $\mathrm{H}=\{H_{1},H_{2},\dots ,H_{n},\dots \}$ and the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ $\mathbb{N}=\{1,2,\dots ,n,\dots \}$ be orthogonal (singular). Then the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ admits a consistent criterion of hypotheses.

The singularity of probability measures implies an existence of the family $\{X_{ik}\}$ of S-measurable sets such that for any $i\ne k$ we have $\mu _{H_{k}}(X_{ik})=0$ and $\mu _{H_{i}}(E-X_{ik})=0$.

Let us consider the sets $X_{i}=\bigcup _{k\ne i}(E-X_{ik})$, then Therefore, $\mu _{H_{i}}(X_{i})=0$; $\mu _{H_{i}}(E-X_{i})=1$. On the other hand, for $k\ne i$ we have $\mu _{H_{k}}(E-X_{i})=0$. This means that the family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ 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 $M_{B}\subset {M}^{\sigma }$ is called a Banach space of measures if:

Let $M_{B}$ be a Banach space of measures, then in $M_{B}$ there exists a family of pairwise orthogonal probability measures $\{\mu _{i},\hspace{0.2778em}i\in I\}$ such that where $M_{B}(\mu _{i})$ is the Banach space of elements ν of the form: with the norm

Let $M_{B}=\bigoplus _{i\in \mathbb{N}}M_{B}(\mu _{H_{i}})$ be a Banach space of measures. The family of probability measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ admits a consistent criteria of hypotheses if and only if the correspondence $f\to l_{f}$ defined by the equality is one-to-one.

Here $l_{f}$ is the linear functional on $M_{B}$, $f\in F(M_{B})$.

Sufficiency. For $f\in F(M_{B})$ we define the linear continuous functional $l_{f}$ by the equality $\int _{E}f(x)\hspace{0.1667em}\mu _{H}dx)=l_{f}(\mu _{H})$. Denote as $I_{f}$ a countable subset in $\mathbb{N}$, for which $\int _{E}f(x)\hspace{0.1667em}\mu _{H_{i}}(dx)=0$ for $i\notin I_{f}$. Let us consider the functional $l_{f_{H_{i}}}$ on $M_{B}(\mu _{H_{i}})$ to which it corresponds. Then for $\mu _{H_{1}},\mu _{H_{2}}\in M_{B}(\mu _{H_{i}})$ we have: Therefore $f_{H_{1}}=f_{1}$ a.e. with respect to the measure $\mu _{H_{i}}$. Let $f_{H_{i}}>0$ a.e. with respect to the measure $\mu _{H_{i}}$ and then Denote $C_{H_{i}}=\{x\mid f_{H_{i}}(x)>0\}$, then $\int _{E}f_{H_{i}}(x)\hspace{0.1667em}\mu _{H_{j}}(dx)=l_{f_{H_{i}}}(\mu _{H_{j}})=0$, $\forall j\in \mathbb{N}$. Hence it follows, that $\mu _{H_{j}}(C_{H_{i}})=0$, $\forall j\in \mathbb{N}$. On the other hand $\mu _{H_{i}}(E-C_{H_{i}})=0$. Therefore the family $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$ is weekly separable and

Let us consider the sets $\overline{C}_{H_{i}}=C_{H_{i}}\diagdown C_{H_{i}}\cap \bigcup _{k\ne i}C_{H_{k}}$. It is obvious that $\{\overline{C}_{H_{i}},i\in \mathbb{N}\}$ is a disjunctive family of S-measurable sets and $\mu _{H_{i}}(C_{H_{i}})=1$, $\forall i\in \mathbb{N}$. Let us define a mapping $\delta :(E,S)\to (\mathrm{H},\mathrm{B}(H))$ like that $\delta (\overline{C}_{H_{i}})=H_{i}$, $\forall i\in \mathbb{N}$. We have $\mu _{H_{i}}(\{x\mid \delta (x)=H_{i}\})=1$, $\forall i\in \mathbb{N}$. Therefore δ is a consistent criterion of hypotheses.

Necessity. Since the family of probability measures $\{\mu _{H_{i}},\hspace{2.5pt}i\in \mathbb{N}\}$ admits a consistent criterion of hypotheses and this family is strongly separable, so there exist S-measurable sets $X_{i}$, $i\in \mathbb{N}$, such that We put the linear continuous functional $l_{X_{i}}$ into the correspondence to a function $I_{X_{i}}\in F(M_{\beta })$ by the formula: We put the linear continuous functional $l_{f_{H_{1}}}$ into the correspondence to the function Then for any $\mu _{H_{2}}\in M_{B}(\mu _{i})$

Let $\varSigma =\{l_{f}\}$ be the collection of extensions of the functional $l_{f_{_{H_{1}}}}:M_{\beta }(\mu _{_{H_{1}}})\to R$ satisfying the condition $l_{f}\le p(x)$ on those subspaces where they are defined.

Let us introduce a partial ordering on Σ having assumed $l_{f_{1}}<l_{f_{2}}$ if $l_{f_{2}}$ is defined on a larger set than $l_{f_{1}}$ and $l_{f_{1}}(\mu )=l_{f_{2}}(\mu )$ where both of them are defined.

Let $\{l_{f_{i}}\}_{i\in I}$ be a linear ordered subset in Σ. Let $M_{B}(\mu _{_{H_{i}}})$ be the subspace on which $l_{_{f_{H_{i}}}}$ is defined. Define $l_{f}$ on $\bigoplus _{i\in I}M_{B}(\mu _{_{H_{i}}})$, having assumed $l_{f}(\mu )=l_{_{f_{H_{i}}}}(\mu )$ if $\mu \in M_{B}(\mu _{_{H_{i}}})$.

It is obvious, that $l_{f_{H_{i}}}<l_{f}$. 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 ${X^{\prime }}$ satisfying the condition $\lambda (x)\le p(x)$ for $x\in {X^{\prime }}$. But ${X^{\prime }}$ must coincide with the entire space $M_{B}$ 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 ${X^{\prime }}=M_{B}$. Therefore the extension of the functional is defined everywhere.

If we put the linear continuous functional $l_{f}$ into the correspondence to the function then we obtain where $\mu _{H}=\sum _{i\in N}\int _{E}g_{H_{i}}\hspace{0.1667em}\mu _{H_{i}}(dx)$. The Theorem 6 is proved.  □

It follows from the proven theorem that the indicated above correspondence puts some functions $f\in F(M_{B})$ into the correspondence to each linear continuous functional $l_{f}$. If in $F(M_{\beta })$ we identify functions coinciding with respect to the measures $\{\mu _{H_{i}},\hspace{0.2778em}i\in \mathbb{N}\}$, then the correspondence will be bijective.

Let $M_{B}=\bigoplus _{H\in \mathrm{H}}M_{B}(\mu _{H})$ be the Banach space of measures, E be the complete separable metric space, S be the Borel σ-algebra in E and $\operatorname{card}\mathrm{H}\le c$. Then in the theory (ZFC) and (MA) the family of probability measures $\{\mu _{H},H\in \mathrm{H}\}$ admits a consistent criteria of hypotheses if and only if the family of probability measures $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ admits an unbiased criterion of any parametric function and the correspondence $f\to l_{f}$ by the equality is one-to-one. Here $l_{f}$ is a linear continuous functional on $M_{B}$, $f\in F(M_{B})$.

Necessity. As the family of probability measures $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ admits a consistent criterion of hypotheses, so the family $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ admits an unbiased criterion of any parametric function and it is strongly separable. So, the family $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ 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 $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$, $\operatorname{card}\mathrm{H}\le c$ is weakly separable. We represent $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ as an inductive sequence $\mu _{H}<\omega _{\alpha }$, where $\omega _{\alpha }$ denotes the first ordinal number of the power of the set H. Since the family $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ is weakly separable, there exists a family of measurable parts $\{X_{H}\}_{H<\omega _{\alpha }}$ of the space E, such that the following relations are fulfilled: for all $H\in [0,\omega _{\alpha })$ and ${H^{\prime }}\in [0,\omega _{\alpha })$.

We define $\omega _{\alpha }$-sequence of parts $B_{H}$ of the space E so that the following relations are fulfilled:

Assume that $B_{0}=X_{0}$. Let further the partial sequence $\{B_{{H^{\prime }}}\}_{{H^{\prime }}<H}$ be already defined for $H<\omega _{\alpha }$.

It is clear, that ${\mu }^{\ast }(\bigcup _{{H^{\prime }}<H}B_{{H^{\prime }}})=0$. Thus there exists a Borel subset $Y_{H}$ of the space E, such that the following relations are valid: $\bigcup _{{H^{\prime }}<H}B_{{H^{\prime }}}\subset Y_{H}$ and $\mu (Y_{H})=0$. Assume $B_{H}=X_{H}\setminus Y_{H}$, thereby the $\omega _{\alpha }$ sequence of $\{B_{H}\}_{H<\omega _{\alpha }}$ disjunctive measurable subsets of the space E is constructed. Therefore $\mu _{H}(B_{H})=1$ for all $H<\omega _{\alpha }$. As a family of probability measures $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ admits an unbiased criterion for any parametric function, so there exists a subspace $G\subset \mathrm{B}(E,S)$, containing $e_{E}$ unit and $\mathrm{B}(E,S)$ can be imagine as a topological sum of G and $H_{0}={\mu }^{-1}(0)$, where the functional and a family $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ is strongly separable, subspace G is a grid towards canonical order on G (see [5]). We assume that $S_{0}$ is a minimal σ-algebra of subalgebra S, all function on G are measurable towards $S_{0}$. Then $G\subset B(E,S_{0})\subset B(E,S)$.

Since a subspace G contains $e_{E}$ and represents a grid, then $G\supset B(E,S)$ and that’s why $G=B(E,S)$.

As family $\{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ represents a dense subspace of $exS_{H}$ ($exS_{H}$ are extreme points of $S_{H}$), so $I_{\mu }$ is an ideal in the set $S_{0}$ which contains zero measured sets for all $\mu \in \{\mu _{H},\hspace{0.2778em}H\in \mathrm{H}\}$ and consists only of an empty set.

Hence there exist such sets $\{A_{H},\hspace{0.2778em}H\in \mathrm{H}\}$ that $\mu _{H}(A_{H})=1$ and $A_{H}\cap A_{{H^{\prime }}}=\varnothing $ for a $H\ne H_{0}$ and $E=\bigcup _{H\in \mathrm{H}}A_{H}$ is a set $S_{0}$. It follows from the condition of this theorem that for every $T\in \mathrm{B}(\mathrm{H})$ in G there exists $f_{T}$ function, which is a consistent criterion of $g_{T}$ parametric function. If $A=\{x\mid f_{t}(x)\ne 0\}$, then $\bigcup _{H\in T}A_{H}\subset A$, $A\cap A_{H}=\varnothing $ for all $H\notin T$ and hence $\bigcup _{H\in T}A_{H}=A$ implying that $\bigcup _{H\in T}A_{H}\subset S_{0}$.

Then, the mapping $\delta (x)=H$ if $x\in A_{H}$ for all $H\in \mathrm{H}$ is a consistent criterion of hypotheses. The theorem is proved.  □