Deduce From the Previous Question That There Exists a Unique a Continuous Function

Unique Continuous Function

Handbook of Dynamical Systems

Roger D. Nussbaum , in Handbook of Dynamical Systems, 2002

3 Linear autonomous FDEs

Suppose that $L:C([-R,0],{ℝ}^n)\to {ℝ}^n$ is a bounded linear map; L extends uniquely in the obvious way to a bounded, complex linear map, which we shall also denote by L, from $C(-R,0),{ℂ}^n)$ to ${ℂ}^n$ . More generally, we assume throughout this section that $L:C([-R,0],{ℂ}^n)$ is a bounded, complex linear map. If $\theta \in Y$ and we identify ${ℂ}^n$ with ${ℝ}^{2n}$ , the results of the previous section prove that there is a unique, continuous function $x:[-R,\infty ]\to {ℂ}^n$ with

(5) $\begin{array}{ll}{x}^{\prime }(t)=L(x_t)\hfill & \text{for}t⩾0\text{and}{x}_0=\theta .\hfill \end{array}$

for each $\theta \in Y,$ let x ( $\cdot ;$ θ):= x denote the corresponding solution of Equation (5) and for $t⩾0$ define T(t): Y → Y by T(t)(θ) = x_t . One can prove easily that for each $t⩾0$ , T(t): Y → Y is a bounded, complex linear map and satisfies the following properties:

(1)

T(0) = I;

(2)

$T\left(t+s\right)=T\left(t\right)T\left(s\right)$ for all $t,s⩾0;$

(3)

${lim}_{t\to s^{+}}T\left(t\right)\left(\theta \right)=T\left(s\right)\left(\theta \right)$ for $s⩾0$ and all $\theta \in Y$ .

In general, a family of bounded linear maps T(t), $t⩾0$ of a Banach space Z into itself is called a C ₀-semigroup if it satisfies properties (1), (2), and (3). In our case, an application of the Ascoli-Arzela theorem shows that T(t) is compact for all $t⩾R$ . In general, a C ₀-semigroup $〈T(t):t⩾0〉$ is called "eventually compact" if there exists $t_0>0$ such that T(t) is compact for all $t_0>t_0.$

There is an extensive literature concerning linear semigroups. We refer to [5,16,67,72,80] for expositions of the general theory and to [9,19] for aspects of the theory for linear, autonomous FDEs. Here we recall only a few facts and refer the reader to [9,19] and the general references for proofs and further details.

In general, if $〈T(t):t⩾0〉$ is a C ₀-semigroup on a Banach space Z, one defines an operator A, called the infinitesimal generator of $〈T(t):t⩾0〉,$ by

(6) $A(\phi )=\psi \iff \underset{t\to 0^{+}}{\text{lim}}\Vert \frac{T(t)\phi -\phi }{t}\Vert =0.$

By definition, D(A), the domain of A, is the set of $\phi \in Z$ for which the limit in Equation (6) exists. One proves that A is a closed, densely defined linear operator. In our case, $D(A)\{\phi \in Y\}\phi$ is continuously differentiable on $[-R,0]$ and ${\phi }^{\prime }(0)=L(\phi )\}$ and $A(\phi )={\phi }^{\prime }$ .

In general, for a closed, densely defined linear operator $A:D(A)\subset Z\to Z$ (Z a complex Banach space), p(A), the resolvent set of A, is the set of $z\in ℂ$ such that zI - A is a one-to-one map of D(A) onto Z. The spectrum of $A,\sigma (A)$ , is the complement of p(A) in $ℂ;$ and ${\sigma }_P(A),$ the point spectrum of A, is the set of $z\in ℂ$ such that $zI-A$ is not one-to-one. In our case, $\sigma (A)={\sigma }_P(A),\sigma (T(t))\backslash \left\{0\right\}={\sigma }_P(T(t))\backslash \left\{0\right\}=\text{exp}(t\sigma (A)),$ and one can give an explicit description of $\sigma (A)$ . For each $z\in ℂ$ and $\upsilon \in {ℂ}^n$ , define $e(z,\upsilon )\in Y$ by

$\begin{array}{ll}e(z,v)(t)=\text{exp}(zt)v,\hfill & -R⩽t⩽0.\hfill \end{array}$

For $z\in ℂ$ , define $\Delta (z):{ℂ}^n\to {ℂ}^n$ by

(7) $\Delta (z)(\upsilon )=z\upsilon -L\left(e(z,\upsilon )\right),$

and note that $\Delta (z)$ is a complex linear map and $\Delta (z)(\upsilon )=0$ if and only if $e(z,\upsilon )\in D(A)$ . One can prove that $z\in \sigma (A)$ if and only if there exists $\upsilon \in {ℂ}^n,v\ne 0,$ with $\Delta (z)(\upsilon )=0$ ; and since $\Delta (z)$ is linear, the latter condition is equivalent to

(8) $det\left(\Delta (z)\right)=0,$

the so-called characteristic equation. If $z\in ℂ$ and $\upsilon \in {ℂ}^n$ satisfy $\Delta (z)(v)=0,$ one easily checks that $y(t):={\mathrm{e}}^{zt}\upsilon$ satisfies $y^{\prime }(t)=L(y_t)$ for all $t\in ℝ.$

For each real number pi, it is relatively easy to prove that the set {z | det(Δ(z)) = 0 and $\text{Re}(z)>\mu \}$ is finite. Thus, for each $\lambda \in \sigma (A)$ , there is a number $r_{\lambda }>0$ and ball $B_{\lambda }=\{w\in ℂ||w-\lambda |⩽r\lambda \}$ such that $B_{\lambda }\cap B_{{\lambda }^{\prime }}=\theta$ whenever $\lambda ,{\lambda }^{\prime }\in \sigma$ (A) and $\lambda \ne {\lambda }^{\prime }$ . Let ${\Gamma }_{\lambda }$ denote the boundary of $B_{\lambda }$ , oriented counterclockwise and, using the functional calculus for linear operators (see [30,80]), define

$P_{\lambda }=\left(\frac{1}{2\pi i}\right){\int }_{{\Gamma }_{\lambda }}{(\zeta I-A)}^{-1}d\zeta .$

It follows from the functional calculus (see [30,80]) that $P_{\lambda }:Y\to Y$ is a bounded linear projection $(\text{so}{P}_{\lambda }^2=P_{\lambda })$ and that $P_{\lambda }{P}_{{\lambda }^{\prime }}=0$ whenever $\lambda ,{\lambda }^{\prime }\in \sigma (A)$ and $\lambda \ne {\lambda }^{\prime }.$ If $M_{\lambda }:=R(P_{\lambda }):=$ the range of $P_{\lambda },M_{\lambda }$ is a closed linear subspace of $Y,M_{\lambda }$ is contained in the domain of A and $A{P}_{\lambda }=P_{\lambda }A$ . Furthermore, λ is a pole of the resolvent operator $\zeta \to {(\zeta -A)}^{-1}$ ; and if the order of this pole is k, then

$M_{\lambda }=\left\{y\in Y|{(\lambda I-A)}^k(y)=0\right\}.$

Indeed, all of the above facts have analogues for a closed, densely defined linear operator A on a general Banach space and a isolated point λ of $\sigma (A)$ : see [30,80]. If Λ is a finite subset of $\sigma (A),$ one can define $P_{\Lambda }={\sum }_{\lambda \in \Lambda }{P}_{\lambda }$ and easily check that $P_{\Lambda }$ is a bounded, linear projection, $M_{\Lambda }:=R(P_{\Lambda })$ is a closed linear subspace of $Y,M_{\Lambda }$ is the direct sum of $M_{\lambda }$ for $\lambda \in \Lambda$ , and $A{P}_{\Lambda }=P_{\Lambda }A.$

In our case A is the infinitesimal generator of an eventually compact C ₀-semigroup $(T(t):t⩾s),$ and one can derive from this that each $M_{\lambda }$ is finite dimensional and $T(t)P_{\lambda }=P_{\lambda }T(t)$ for all $t⩾0$ . A deeper fact is that the dimension of $M_{\lambda }$ is the multiplicity of λ as a zero of the equation det $(\Delta (z))=0:$ see [9,19,26,36]. Of course it is desirable to have explicit formulas for $P_{\lambda }(y)$ , and the reader is referred to [9,19] for such formulas.

Just the knowledge that $\lambda \in \sigma (A)$ is an isolated point of $\sigma (A)$ implies that $M_{\lambda }$ is contained in D(A), the domain of A, and $A{P}_{\lambda }=P_{\lambda }A;$ so $A_{\lambda }:=A|M_{\lambda }$ is a closed operator defined on all of $M_{\lambda }$ , and the closed graph theorem implies that $A_{\lambda }:M_{\lambda }\to M_{\lambda }$ is a bounded linear operator. It follows that $T_{\lambda }(t):=T(t)|M_{\lambda }$ is given by $T_{\lambda }(t)=exp(t{A}_{\lambda })$ for $t⩾0$ and the semigroup $(T_{\lambda }(t):t⩾0)$ extends to a C ₀-group given by exp $(t{A}_{\lambda })$ for $t\in ℝ$ .

If $\omega \in ℝ$ and $\Lambda =\{\lambda \in \sigma (A)|\text{Re}(\lambda )⩾\omega \}$ and $R(I-P_{\Lambda })$ denotes the range of $I-P_{\Lambda }$ , one can prove that $T(t)|R(I-P_{\Lambda })$ is a C ₀-semigroup with infinitesimal generator $A|R(I-P_{\Lambda })$ . Furthermore, there is a constant M such that $\left|\right|T(t)(I-P_{\Lambda }||⩽M\text{exp}(\omega t)$ for all $t⩾0$ x. Using $P_{\Lambda }$ , one can write

$T(t)y=T(t)(I-P_{\Lambda })y+T(t)P_{\Lambda }y.$

One has control of the growth of $T(t)(I-P_{\Lambda })$ as $t\to \infty ,$ and $T(t)P_{\Lambda }$ can be explicitly computed.

The above remarks provide a sketchy introduction to the theory of linear autonomous FDEs. There remain intriguing questions which we have not mentioned. For example, do there exist nonzero solutions of Equation (5) which converge to zero at a "super-exponential rate" as $t\to \infty$ ? The reader is referred to [19] for further details and references to the literature.

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Handbook of Computability Theory

Helmut Schwichtenberg , in Studies in Logic and the Foundations of Mathematics, 1999

4.7 Cartesian Products

We define an information system A × B = (C, Con, ├) from A and B in such a way that the inclusion ordering ⊆ on the objects of |A × B| is isomorphic to the Cartesian product of the inclusion orderings on |A| and |B|. Without loss of generality we may assume that A and B are disjoint. But then any pair ( x, y) of elements x ∈ |A| and y ∈ |B| can be approximated in each component separately. Hence we choose as tokens in A × B simply the union C := A $\cup ︀$ B. Consistency and entailment is inherited in the expected way from A and B:

$\begin{array}{l}X\in \text{CON:}\equiv \text{X}\cap \text{A}\in {\text{CON}}_{A}andX\cap B\in {\text{CON}}_B,\\ X⊢c:\equiv \left(c\in A\Rightarrow X\cap A{⊢}_{A}c\right)\text{and}\left(c\in B\Rightarrow X\cap B{⊢}_{B}c\right).\end{array}$

It is then obvious that we have

Lemma

If A and B are information systems with A ∩ B = Ø, then so is A × B defined as above. The objects of A × B are exactly the unions x $\cup ︀$ y of the objects of x ∈ |A| and of y ∈ |B|.

When using both → and × to build information systems, we assume that × has a higher precedence that →, hence, e.g., $\text{A}\times \text{B}\to \text{}\text{C\hspace{0.17em}means}\left(\text{A}\times \text{B}\right)\to C$ .

Remark

Clearly the pairs (x, y) ∈ |A| × |B| are in a natural bijective correspondence with the objects x $\cup ︀$ y of |A × B|. Therefore from now on we will usually write x, y for x $\cup ︀$ y, to increase readability. It should be clear from the context where a comma needs to be replaced by a union.

Clearly the projections π_i : A ₁ × A ₂ → A _i defined by π_i (x ₁, x ₂) := x_i for i = 1, 2 are continuous.

Lemma

(Universal property of Cartesian products). Let A, B and C be information systems such that A ∩ B = Ø. Then for any pair f: C→A and g: C→ B of continuous functions there is a unique continuous function h: C→A × B such that f = π ₁ $\circ$ h and g = π ₁ $\circ$ h.

Proof

Uniqueness follows from the fact that any such h must satisfy h(x) = (f(x), g(x)) for x ∈ |C|. To prove existence, define h(x) := (f(x), g(x)) for x ∈ |C|. We must show that this h is continuous. Monotonicity is obvious; for PFS assume b ∈ (f(x), g(x)). Since A ∩ B = ∅, the token b must be in exactly one component, say B. So b ∈ g(x), hence b ∈ g( $\overline{X}$ ) for some X ⊆^fin x and therefore b ∈ h( $\overline{X}$ ) = (f( $\overline{X}$ ), g( $\overline{X}$ )).

As an application let us construct the product f × g of two continuous functions f: A→C and g: B→D (with A ∩ B = Ø).

f × g satisfies (f × g)(x, y) = (f(x), g(y)) for x ∈ |A| and y ∈ |B|.

Lemma

Let A, B and C be information systems with A ∩ B = Ø. A function f: A × B→C is continuous iff it is continuous in each argument separately, i.e. all its sections $f_1^y:\text{A}\to \text{C},f_1^y\left(x\right):=f\left(x,y\right)$ for fixed y ∈ |B| and similarly $f_2^x:\text{B}\to \text{C},$ $f_2^x\left(y\right):=f\left(x,y\right)$ for fixed x ∈ |A| are continuous.

Proof

The function sending x to (x, y) for fixed y clearly is continuous, hence by composition so is $f_1^y$ for $f_2^x$ the argument is similar. Conversely, assume that all $f_1^y$ and $f_2^x$ are continuous. To prove monotonicity of f, assume x ⊆ x′ and y ⊆ y′ with x, x′ ∈ |A| and y, y′ ∈ |B|. Then f(x, y) ⊆ f(x′, y) ⊆ f(x′, y′) by monotonicity of the sections. To prove PFS for f, assume c ∈ f(x, y). Then $c\in f_1^x\left(y\right)$ , hence

$c\in f_1^x\left(\overline{\text{Y}}\right)=f\left(x,\overline{Y}\right)=f_2^{\overline{Y}}\left(x\right)$

for some Y ⊆^fin y by PFS for $f_1^x$ , and also

$c\in f_2^{\overline{Y}}\left(\overline{X}\right)=f\left(\overline{X},\overline{Y}\right)=f\left(\overline{X\cup ︀Y}\right)$

for some X ⊆^fin x by PFS for $f_2^{\overline{Y}}.$

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Theory of Linear Operations

In North-Holland Mathematical Library, 1987

§4 Some applications to the theory of orthogonal expansions.

Theorem 7.

Suppose that the sequences (x_i ), (f_i ) and (y_i ),(ϕ _i ) are biorthogonal and that the equations f_i (x) = ϕ _i (y), for i = 1,2, …, have exactly one solution y = U(x) for every x. Then the convergence of the series ${\displaystyle \sum _{i=1}^{\infty }{h}_i}{x}_i$ implies that of the series ${\displaystyle \sum _{i=1}^{\infty }{h}_i}{y}_i$ for every sequence of numbers (h_i ).

Proof. It is easily seen that if $\underset{n\to \infty }{\lim }{x}_n=x_0$ and $\underset{n\to \infty }{\lim }{y}_n=y_0$ , where y_n = U(x_n ), then y ₀ = U(x ₀). It follows from theorem 7 (Chapter III, §3) p. 26, that the operator U is bounded and linear. Therefore putting ||U|| = M, we have ||U(x)|| ≦ M||x|| and since, by definition, U(x_i ) = y_i for i = 1,2, …, it follows that

$U\left({\displaystyle \sum _{i=1}^n{h}_i{x}_i}\right)={\displaystyle \sum _{i=1}^n{h}_i{y}_i}$

for any real numbers h_i , from which the result immediately follows.

Corollary.

Suppose that (x_i (t)) and (y_i (t)) are orthonormal sequences of continuous functions, and that for every continuous function x(t) there exists a unique continuous function y(t) such that

${\displaystyle \underset{0}{\overset{1}{\int }}{x}_i(t)x(t)dt}={\displaystyle \underset{0}{\overset{2}{\int }}{y}_i(t)y(t)dt}.$

Then if the series ${\displaystyle \sum _{i=1}^{\infty }{h}_i}{x}_i\left(t\right)$ is uniformly convergent so is the series ${\displaystyle \sum _{i=1}^{\infty }{h}_i}{y}_i\left(t\right).$

Analogous corollaries hold for other function spaces.

Theorem 8.

Let (x_i ), (f_i ) be a biorthogonal sequence, where (f_i ) is a total sequence, and let (h_i ) be a sequence of numbers such that whenever (α_i ) is the sequence of coefficients of an element x (i.e. α_i = f_i (x) for i = 1,2, …), (h_iα_i ) is the coefficient sequence of an element y.

If under these conditions, (β_i ) is the coefficient sequence of a bounded linear functional F (i.e. β_i = F(x_i ) for i = 1,2, …), the sequence (h_iβ_i ) is also the coefficient sequence of some bounded linear functional ϕ.

Proof. By hypothesis, the system of equations h_if_i (x) = f_i (y) for i = 1,2, … has, for every x, exactly one solution, which we denote by y = U(x).

The equalities $\underset{n\to \infty }{\lim }{x}_n=x_0$ and $\underset{n\to \infty }{\lim }{y}_n=y_0$ where y_n = U(x_n ) clearly imply that y ₀ = U(x ₀). Consequently, by theorem 7 (Chapter III, §3) p. 26, U is a bounded linear operator. In particular, it is easily checked that

(9) $\begin{array}{lll}U(x_i)=h_i{x}_i\hfill & \text{for\hspace{0.17em}every}\hfill & i=1,2,...\hfill \end{array}$

Now, given a bounded linear functional F such that β_i = F(x_i ) for i = 1,2, …, we have, in view of (9), F[U(x_i )] = h_iF(x_i ) = h_iβ_i , i.e. the numbers h_iβ_i are the coefficients of the functional ϕ = U ^*(F), q.e.d.

Note that if $x=\underset{i\to \infty }{\lim }{x}_i$ U(x) is, by (9), the limit of a linear combination of terms of the sequence (x_i ).

As an easy application of this remark we obtain the following

Theorem 9.

Let (x_i (t)) be an orthonormal sequence of continuous functions which is also a closed sequence in the space C.

If the sequence of scalars (h_i ) transforms every sequence (α_i ) of coefficients of a bounded function into the coefficient sequence (h_iα_i ) of another bounded function, then it transforms every coefficient sequence (β_i ) of any continuous function into the coefficient sequence (h_iβ_i ) of another continuous function.

The converse theorem is also true.

Lastly, we have

Theorem 10.

Let (x_i (t)) be a complete orthonormal sequence of bounded functions in L ^p/(p−1), where p > 1.

If the sequence of scalars (h_i ) transforms the coefficient sequence (α_i ) of an arbitrary function x(t) ∈ L^p into the coefficient sequence (h_iα_i ) of another function y(t) ∈ L^p, then it also transforms every coefficient sequence (β_i ) of an arbitrary function $\overline{x}\left(t\right)\in L^{p/\left(p-1\right)}$ into the coefficient sequence (h_iβ_i ) of a function $\overline{y}\left(t\right)\in L^{p/\left(p-1\right)}$ . If p = ∞, then L^p = M.

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Liftings

Werner Strauss , ... Kazimierz Musiat , in Handbook of Measure Theory, 2002

THEOREM 4.1

For given c.l.d. measure spaces (Ω, ∑, μ) with $\delta \in \vartheta \left(\mu \right)$ and $\upsilon \in \Lambda \left(\mu \right)$ we have the following results.

(i)

$t_{\delta }\subseteq {\tau }_{\delta }\subseteq \sum .$

(ii)

${\tau }_{\delta }\cap {\sum }_0=\varnothing$ and for all A ∈ ∑ there exists a $G\in {\tau }_{\delta }$ with $A\Delta G\in {\sum }_0$ .

(iii)

A subset $K\subseteq \Omega$ is of the 1st category with respect to the topology ${\tau }_{\delta }$ if and only if K is closed and nowhere dense or equivalently if $K\in {\sum }_0$ .

(iv)

The topologies t_ρ and τ_δ are extremally disconnected and $t_{\rho }\subseteq {\tau }_{\rho }$ .

(v)

The topology t_ρ (hence τ_ρ) is Hausdorff if the set $\left\{\rho \left(A\right):A\in \sum \right\}$ separates the points of Ω.

(vi)

$C_{\overline{ℝ}}\left(\Omega ,t_{\rho }\right)=C_{\overline{ℝ}}\left(\Omega ,{\tau }_{\rho }\right)=\left\{{\rho }^0\left(f\right):f\in {\overline{ℒ}}^0\left(\mu \right)\right\}$ and ${\rho }^0\left(f\right)$ is the unique continuous function with respect to t _ρ (respectively, τ_ρ) in $f^{•}$ .

(vii)

$C_b\left(\Omega ,t_{\rho }\right)=C_b\left(\Omega ,{\tau }_{\rho }\right)=\left\{{\rho }^{\infty }\left(f\right):f\in {ℒ}^{\infty }\left(\mu \right)\right\}.$

(viii)

if t_ρ is Hausdorff then the Stone space of the measure algebra ∑/∑₀ is the Stone Cech compactifcation of (Ω, t_ρ).

If T ∈ {t_ρ,τ_ρ} we can add the following conditions:

(ix)

We have $c{l}_T\left(A\right)=\rho \left(A\right)$ for all A ∈ T and $\sum =B\left(\Omega ,{\tau }_{\rho }\right)$ .

(x)

ρ is the unique T-strong lifting for the topological measure space (ρ, T, ∑, μ).

(xi)

The topological measure space (Ω, T, ∑, μ) is a quasi-Radon measure space.

Here (i) follows from Theorem 3.9 and (ii) and (iii) are from A. Ionescu Tulcea (1967a). A proof for (iv) to (viii) is available in A. and C. Ionescu Tulcea (1969a, Chapter V, section 3), as well as the equation $C_{\overline{ℝ}}\left(\Omega ,t_{\rho }\right)=C_{\overline{ℝ}}\left(\Omega ,{\tau }_{\rho }\right)$ in (v).

If δ is the Lebesgue density (see section 2) of the Lebesgue measure space (Ω, ∑, μ) on ℝ and ρ is a lifting with $\delta \left(A\right)\subseteq \rho \left(A\right)$ for A ∈ ∑ then τ_ρ is completely regular and t_ρ = τ_ρ by A. and C. Ionescu Tulcea (1969a, Chapter V, section 4, Theorem 2). But there exists a ${\rho }_0\in \Lambda \left(\mu \right)$ such that ${\tau }_{{\rho }_0}$ is finer than the euclidean topology on ℝ and $t_{{\rho }_0}\ne {\tau }_{{\rho }_0}$ . For any $\lambda \in \Lambda \left(\mu \right)$ with ${\tau }_{\lambda }$ finer than the euclidean topology on ℝ the topology ${\tau }_{\lambda }$ is not normal and every ${\tau }_{\lambda }$ -compact $K\subseteq ℝ$ is finite by A. Ionescu Tulcea (1967a).

It is now easy to see that generally the topological measure space (Ω, T, ∑, μ) from (vii) will not be a Radon measure space. Take for instance as (Ω, ∑, μ) the Lebesgue measure space with $T:={\tau }_{\rho }$ for a lifting $\rho \in \Lambda \left(\mu \right)$ for which ${\tau }_{\rho }$ is finer than the euclidean topology ε_d on ℝ. Since by the above remark any ${\tau }_{\rho }$ -compact subset K of ℝ is finite the inner regularity $\mu \left(A\right)=\sup \left\{\mu \left(K\right):K\subseteq A,K{\tau }_{\rho }-\text{compact}\right\}$ with respect to ${\tau }_{\rho }$ -compact subsets K of ℝ can't be true if μ(A) > 0.

If $\left(\Omega ,\sum ,\mu \right)$ and $\left(\Theta ,T,v\right)$ are measure spaces, $\delta \in \vartheta \left(\mu \right)$ , and $\zeta \in \vartheta \left(v\right)$ then a ∑−T-measurable map $f:\Omega \to \Theta$ is ${\tau }_{\delta }\infty {\tau }_{\zeta }$ -continuous if and only if $f^{-1}\left(\zeta \left(B\right)\right)\subseteq \delta \left(f^{-1}\left(\zeta \left(B\right)\right)\right)$ for all B ∈ T. If moreover $\delta \in \Lambda \left(\mu \right)$ , and $\zeta \in \Lambda \left(v\right)$ these conditions are equivalent with $f^{-1}\left(\zeta \left(B\right)\right)=\delta \left(f^{-1}\left(B\right)\right)$ for all B ∈ T as well as with $\zeta \left(h\right)\circ f=\delta \left(h\circ f\right)$ for all $h\in {ℒ}^{\infty }\left(\mu \right)$ .

Proposition 4.2

If $\left(\Omega ,T,\sum ,\mu \right)$ and $\left(\Theta ,S,T,v\right)$ are topological measure spaces, the map f is a T-S-continuous surjection, and f(δ) exists for $\delta \in \vartheta \left(\mu \right)$ , then f(δ) is T-strong provided δ is T-strong.

But note that under the assumptions of Proposition 4.2 for a strong lifting ζ for v an inverse lifting $\delta \in f^{-1}\left(\zeta \right)$ needs not to be strong, see the example before Theorem 6.20.

As we state below, the last theorem gives in fact a topological characterization for c.l.d. measure spaces having a lifting and there are again characterizations by weaker types of "liftings" in an abounding number, hence we can mention only the most spectacular ones. We call a map φ from ${ℒ}^{\infty }\left(\mu \right)$ into itself satisfying the properties (11), (12) a monotonous lifting if in addition $\phi \left(f\right)⩽\phi \left(g\right)$ for $f,g\in {ℒ}^{\infty }\left(\mu \right)$ and $f⩽g$ , it is called a bounded linear lifting if φ is a linear map with $\Vert \phi \Vert :=\sup \left\{\Vert \phi \left(f\right)\Vert /{\Vert f\Vert }_{\infty }:0<{\Vert f\Vert }_{\infty }<\infty \right\}\omega \infty$ , where $\Vert f\Vert$ is the strict supremum and ${\Vert f\Vert }_{\infty }$ the essential supremum of a function $f\in {ℒ}^{\infty }\left(\mu \right)$ , it is called a function lower respectively upper density if $\phi \left(f\Lambda g\right)=\phi \left(f\right)\Lambda \phi \left(g\right)$ and $\phi \left(f\vee g\right)=\phi \left(f\right)\vee \phi \left(g\right)$ , respectively for $f,g\in {ℒ}^{\infty }\left(\mu \right)$ .

THEOREM 4.3

For given c.l.d. measure spaces (Ω, ∑, μ) the following conditions are all equivalent with the existence of a lifting for ${ℒ}^{\infty }\left(\mu \right)$ .

(i)

There exists a topology $T\subseteq \sum$ such that the topological measure space $\left(\Omega ,T,\sum ,\mu \right)$ is a category measure space.

(ii)

There exists a topology $T\subseteq \sum$ such that $T\cap {\sum }_0=\varnothing$ and a set $K\subseteq \Omega$ is of first category if and only if it is closed and nowhere dense in Ω.

(iii)

There exists a topology $T\subseteq \sum$ such that card $\left(f^{•}\cap C_b\left(\Omega \right)\right)=1$ for any $f\in {ℒ}^{\infty }\left(\mu \right)$ , where $C_b\left(\Omega \right)$ denotes the space of all T-continuous, bounded real-valued functions on Ω.

(iv)

There exists a monotonous lifting for ${ℒ}^{\infty }\left(\mu \right)$ .

(v)

There exists a function lower respectively upper density for ${ℒ}^{\infty }\left(\mu \right)$ .

(vi)

There exists a linear lifting for ${ℒ}^{\infty }\left(\mu \right)$ .

(vii)

There exists a bounded linear lifting φ for ${ℒ}^{\infty }\left(\mu \right)$ of norm $\Vert \phi \Vert \omega 3$ .

For (i) and (ii) compare Graf (1973), for the sophisticated equivalence proof for (vii) see Erben (1983), where an example is given that the bound 3 cannot be improved.

We call $\delta \in \vartheta \left(\mu \right)$ (and $\psi \in \mathcal{G}\left(\mu \right)$ ) almost T-strong, if there exists a set $N\in {\sum }_0$ such that for all $G\in T\cap \sum$ follows $G\backslash N\subseteq \delta \left(G\right)$ (respectively $\psi \left(f\right)\left|N^c=f\right|N^c$ for all $f\in C_b\left(\Omega \right)\cap {ℒ}^{\infty }\left(\mu \right)$ ). δ and ψ are called T-strong in case $N=\varnothing$ . It is obvious that $\delta \in \vartheta \left(\mu \right)$ is T-strong if and only if $T\subseteq {\tau }_{\delta }$ .

If ε_d is the euclidean topology of ℝ ^d then it is well known that the Lebesgue density D and any lifting ρ for the Lebesgue measure with $D\left(A\right)\subseteq \rho \left(A\right)$ for $A\in \sum$ are ε_d -strong, and so are the linear liftings ψ₁, ψ₂ from section 3 obtained by L. Fejér's theorem (see Hoffmann (1965, pages 20 and 33)). Any hyperstonian space has a uniquely determined strong lifting, which is given by choosing the unique continuous function from each equivalence class of $L^{\infty }\left(\mu \right)$ . For any complete measure space $\left(\Omega ,\sum ,\mu \right)$ any $\rho \in \Lambda \left(\mu \right)$ is t_ρ -strong as well as τ_ρ -strong by the definition of t_ρ and τ_ρ .

Proposition 4.4

If the topology T is $T_{3\frac{1}{2}}$ then for each $\rho \in \Lambda \left(\mu \right)$ the following conditions are equivalent.

(i)

ρ is almost T-strong.

(ii)

ρ^∞ is almost T-strong.

(iii)

There exists a $N\in {\sum }_0$ such that $\rho \left(F\right)\subseteq F\cup N$ for all closed $F\in \sum$ .

(iv)

There exists a $N\in {\sum }_0$ such that ${\rho }^0\left(f\right)\left|N^c=f\right|N^c$ for all $f\in \overline{C}\left(\Omega \right)\cap {ℒ}^0\left(\mu \right)$ , where $\overline{C}\left(\Omega \right)$ denotes the space of all continuous functions from Ω into $\overline{ℝ}$ .

(v)

There exists a $N\in {\sum }_0$ such that ${\rho }^{\infty }\left(f\right)\left|N^c=f\right|N^c$ for all $f\in C_b\left(\Omega \right)\cap {ℒ}^{\infty }\left(\mu \right)$ , where $C_b\left(\Omega \right)$ denotes the space of all bounded continuous functions from Ω into ℝ.

Here we can choose the same set $N\in {\sum }_0$ in (i) to (iv), where in particular $N=\varnothing$ for T-strong ρ and remember that ρ, ρ^∞, and ρ⁰ are in biunique correspondence by means of the equations ${\rho }^{\infty }\left({\chi }_A\right)={\rho }^0\left({\chi }_A\right)={\chi }_{\rho \left(A\right)}$ for $A\in \sum$ .

The implication (i) ⇒ (iv) is quickly achieved by observing ${\rho }^0={\rho }_0$ if ${\rho }_0\left(f\right)=\sup \left\{r\in \mathbb{Q}:\omega \in \rho \left(\left\{f>r\right\}\right)\right\}$ since then ${\rho }_0\left(f\right)\left|N^c\ge f\right|N^c$ for $f\in \overline{C}\left(\Omega \right)$ and ${\rho }_0\left(-f\right)=-{\rho }_0\left(f\right)$ hence ${\rho }_0\left(f\right)\left|N^c\ge f\right|N^c$ if if ρ is almost T-strong with universal set $N\in {\sum }_0$ . The implication (iv) ⇒ (ii) and the equivalence of (i) and (ii) are obvious.

Moreover (ii) ⇒ (i) works for almost T-strong $\psi \in \mathcal{G}\left(\mu \right)$ , since for a $T_{3\frac{1}{2}}$ topology T follows ${\chi }_G=\sup \left\{h\in C_b\left(\Omega \right):h⩽{\chi }_G\right\}$ if $G\in T$ hence ${\chi }_{G\backslash N}⩽\psi \left({\chi }_G\right)|N^c$ therefore $G\backslash N\subseteq \underset{\_}{\psi }\left(G\right)$ for $G\in \sum \cap T$ if $\underset{\_}{\psi }$ is defined by $\underset{\_}{\psi }\left(A\right)=\left\{\psi \left({\chi }_A\right)=1\right\}$ for $A\in \sum$ , where for $\rho \in \Lambda \left(\mu \right)$ follows $\underset{\_}{\rho }=\rho$ . So we have in addition the following result.

Proposition 4.5

If the topology T is $T_{3\frac{1}{2}}$ and the measure space $\left(\Omega ,\sum ,\mu \right)$ is complete then the existence of a T-almost strong density for μ is equivalent with the existence of a T-almost strong (linear) lifting. Here we may replace "almost strong" by "strong".

If a topological measure space admits a T-strong density its measure has to be of full support, i.e., $\sup p\left(\mu \right)=\Omega$ , since then for all G ∈ T follows $G=\varnothing$ from $G={\sum }_0$ . The notion of the almost strong lifting allows us to cover in full generality the cases with $\sup p\left(\mu \right)\ne \Omega$ , for which the notion of strong lifting is inappropriate.

Proposition 4.6

if (Ω, ∑, μ) is a complete measure space with $\sup p\left(\mu \right)=\Omega$ , then from the existence of a T-almost strong lifting for μ follows the existence of a T-strong lifting.

This is easily achieved for $\rho \in \Lambda \left(\mu \right)$ with $G\backslash N\subseteq \rho \left(G\right)$ for $G\in T\cap \sum ,N\in {\sum }_0$ by choosing an ultrafilter U(ω) finer than the filter basis $\left\{G\in T:\omega \in G\right\}$ for $\omega \in N$ and putting

$\begin{array}{ll}\overline{\rho }\left(A\right):=\left(\rho \left(A\right)\cap N^c\right)\cup \left\{\omega \in N:\mathcal{U}\left(\omega \right)\right\}\hfill & \text{for}A\in \sum \hfill \end{array}.$

The space (Ω, T, ∑, μ) (or just μ) has the almost strong lifting property (ASLP for short), if there exists an almost T-strong lifting for μ. Since for general spaces (Ω, T, ∑, μ) there is no "natural" candidate for an almost strong lifting, one possible issue is to check whether arbitrary liftings are almost strong. This leads to the stronger notion of universal strong lifting property (USLP for short), which says that $\Lambda \left(\mu \right)\ne \varnothing$ and every $\rho \in \Lambda \left(\mu \right)$ is almost T-strong. Results sufficient for applications (polish spaces) rely on the next result which is a generalization of a result from Maher (1978), see also Fremlin (2000), and Macheras and Strauss (1996a). Before we state it we need a modification of the purely topological notion of network (see Gruenhage (1984)). A family $F\subseteq \sum$ is called a measurable network for the pretopological measure space (Ω, T, ∑, μ) if for each G ∈ T there exists a subfamily $\mathcal{G}\subseteq F$ such that $G=\cup \mathcal{G};$ we denote by $mn\omega \left(\mu \right)$ the least cardinal of a measurable network for (Ω, T, ∑, μ).

LEMMA 4.7

If (Ω, T, ∑, μ) is a pretopological measure space with a measurable network F then $\rho \in \Lambda \left(\mu \right)$ is almost T-strong if there exists $N\in {\sum }_0$ such that $G\backslash N\subseteq \rho \left(G\right)$ for all G ∈ F.

The last lemma remains true if we replace liftings by monotonous liftings.

It follows that a complete topological measure space (Ω, T, ∑, μ) has the USLP if there exists a cardinal ℵ with $mnw\left(\mu \right)<\aleph$ and if for any family $A\subseteq \sum$ with card $\left(A\right)<\aleph$ and $\mu \left(A\right)=0$ for $A\in A$ follows $\cup A\in \sum$ and $\mu \left(\cup A\right)=0$ . Indeed for given lifting $\rho \in \Lambda \left(\mu \right)$ apply Lemma 4.7 for $N:={\displaystyle {\cup }_{F\in F}\left(F\backslash \rho \left(F\right)\right)}$ if F is a measurable network of cardinality $mnw\left(\mu \right)$ . For $\aleph ={\aleph }_1$ this gives the next result of Fremlin (200?, 453F).

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Probability on MV-Algebras

Beloslav Rieĕan , Daniele Mundici , in Handbook of Measure Theory, 2002

DEFINITION 2.1

Let M be a σ-complete MV-algebra. A state is a map m : M → [0, 1] satisfying the following conditions for all a, b, c, a_n, b_n ∈ M :

(i)

m(1) = 1;

(ii)

whenever b,c ∈ M and c ⊙ b=0 it follows that m(b⊕ c)=m(b)+m(c);

(iii)

if a_n ↗a, then m(a_n) ↗ m(a).

We say that m is faithful ⁵ if m(x) ≠ 0 whenever 0 ≠ x ∈ M.

The assumption c ⊙ b=0 in condition (ii) amounts to asking b⊕c = b + c, where + is addition in the ℓ-group with strong unit corresponding to M, as given in Theorem 1.7 and (4).

Putting together Theorems 1.16 and 1.14 we have the following integral representation theorem, where о denotes composition of functions:

PROPOSITION 2.2

Let M be a σ-complete MV-algebra equipped with a state m. Let Ω = M(M) be the space of maximal ideals of M with the spectral topology. Let the tribe (Ω, F) and the σ-homomorphism η:F → M be as in Theorem 1.16. Then (Ω, F) is a tribe and m о η is a state on it. Let (Ω, S, P) be as given by Theorem 1.14 starting from the triplet (Ω, F, m о η). Let the map^* : M → F send each a ∈ M into the unique continuous function a ^* ∈ F such that η. Then for all a ∈ M,

Generalizing Carathéodory's approach to boolean probability theory, we shall be mainly concerned with a situation where the state m is faithful, as follows:

DEFINITION 2.3

A probability MV-algebra is a pair (M, m) where M is a σ-complete MV-algebra and m is a faithful state on M. We say that (M, m) is divisible if so is M. We say that (M,m) is weakly σ-distributive if so is M.

EXAMPLE

Let (Ω, F) be a tribe with a state m. Let J be the set of all f ∈ F such that m(f) = 0. Then J is an ideal of the MV-algebra F, and J is closed under countable suprema. The quotient MV-algebra M = F|J is σ-complete, ⁶ and the quotient map θ: F → M is a σ-homomorphism. Two functions f,g ∈ F have the same image under $\theta \text{iff\hspace{0.17em}m}\left(\left|f-g\right|\right)=0.$ Let the map $\stackrel{⌣}{m}:M\to \left[0,1\right]$ be defined by $\stackrel{⌣}{m}\left(\theta \left(f\right)\right)=m\left(f\right)$ for each element θ(f) ∈ M. Then the map is well defined and is a faithful state on M, and (M,m) is a probability MV-algebra.

A variant of Proposition 2.2 can be used to show that this construction yields the most general possible example of a probability MV-algebra.

A particular case of this example is given by the real unit interval [0, 1] equipped with the state m(x) = x.

As another particular case, let E be a non-empty, finite or denumerably infinite, set of elements e ₁, e ₂, …, equipped with a corresponding sequence p ₁, p ₂ … of real numbers p _i > 0 such that ${\displaystyle {\sum }_i{p}_i=1.}$ Let M be the set of all sequences f : E → [0, 1] equipped

with termwise MV-operations. Let m : M → [0, 1] be defined by $m\left(f\right)={\displaystyle {\sum }_i{f}_{i}pi}$ Then m is a faithful state on the complete MV-algebra M.

A first property of probability MV-algebras is given by the following

PROPOSITION 2.4

Any probability MV-algebra (M, m) is weakly σ-distributive.

PROOF

Let b_ij ∈ M be a double sequence having the property that, for each i ∈ ℕ, $b_{ij}⩾b_{ij+1}and{\bigwedge }_j{b}_{ij}=0,$ with the intent of proving $0={\bigwedge }_{\varphi \in {\mathbb{N}}^{\mathbb{N}}}{\bigvee }_i{b}_{i.\varphi \left(i\right).}$ Assume (absurdum hypothesis) 0< b∈ M is a lower bound for each element ${\bigvee }_i{b}_{i.\varphi \left(i\right).}$ , where ϕ ranges over ℕ^ℕ. From the assumed faithfulness of m it follows that m(b) > 0. Since for each fixed i,b ∧ b_ij tends to 0 as j tends to infinity, then so does m(b ∧b_ij . There is j^* = ψ(i) such that $m\left(b\bigwedge b_{i\psi \left(i\right)}\right)⩽m\left(b\right)/2^{i+1}$ . By absurdum hypothesis, $b⩽{\bigvee }_i{b}_{i.\psi \left(i\right)}.$ Then from the distributivity law (2), we get the contradiction $m\left(b\right)=m\left(b\wedge {\bigvee }_i{b}_{i.\psi \left(i\right)}\right)=m\left({\bigvee }_i\left(b\wedge b_{i.\psi \left(i\right)}\right)\right)⩽{\displaystyle {\sum }_{i}m\left(b\wedge b_{i.\psi \left(i\right)}\right)<m\left(b\right).}$

As usual, we let ℝ denote the set of real numbers. In the point-free version of probability, the algebraic counterpart of a random variable is not given by a real-valued function over the sample space Ω but, vice versa, by a function from the σ-algebra B(ℝ) of Borel sets in ℝ into the σ-boolean algebra of events. The explicit construction of sums and products of such random variables in terms of boolean operations was successfully carried out by Carathéodory. In order to generalize his results to MV-algebras we prepare

DEFINITION 2.5

Let M be a σ-complete MV-algebra. An n-dimensional observable of M is a map x:B(ℝⁿ) → M satisfying the following conditions:

(i)

x(ℝⁿ)=1;

(ii)

whenever A,B ∈ B(ℝⁿ) and A ∩ B = Ø, then x(A) ⊙ x(B)=0 and x(A ∪ B)= x(A)⊕ x(B)

(iii)

for all A, A ₁, A ₂, … ∈ B (ℝⁿ), if A_n ↗ A, then x(A_n) ↗ x(A).

When n = 1 we simply say that x is an observable.

Thus, condition (ii) above states that, whenever A ∩B=0 then x(A∪B) = x(A)+x(B) in the ℓ-group with strong unit corresponding to M via Theorem 1.7.

REMARK

In the particular case when M is a σ-complete boolean algebra, an n-dimensional observable x:B(ℝⁿ) → M is a σ -homomorphism into M of the σ-complete boolean algebra B(ℝⁿ) of all Borel sets in (ℝ ⁿ ). In general, this need not be true of σ-complete MV-algebras.

EXAMPLES

1

Let (Ω, S, P) be a probability space, ξ:Ω → ℝ a random variable, and F the full tribe of all [0, 1]-valued S-measurable functions on Ω. Let the map x:B(ℝ) → M be defined by $x\left(A\right)={\chi }_{{\xi }^{-1}\left(A\right)}$ , where χE is the characteristic function of E ⊆ Ω. Then x is an observable.

2

Let M = [0, 1] with the usual MV-algebraic operations. Let ϕ:ℝ →[0,∞[be a (Borel measurable) density function, in the sense that ${\displaystyle {\int }_{-\infty }^{\infty }\varphi \left(t\right)\text{d}t=1.}$ Define x:B(ℝ) → M by the formula $x\left(A\right)={\displaystyle {\int }_A\varphi \text{d}\lambda },$ where λ is Lebesgue measure. Then x is an observable of M.

3

Again assume M=[0,1]. Let υ₁, … υ_m, be a finite set of real numbers. Let (p ₁, …, p_m) be an m-tuple of real numbers ⩾ 0 such that p₁ + … + p_m = 1. Let y:B(ℝ) → M be defined by $y\left(A\right)={\displaystyle {\sum }_{{\upsilon }_i\in A}{P}_i}.$ Then y is an observable of M.

4

Given a σ-complete MV-algebra M, let G be the ℓ-group with strong unit u = 1 corresponding to M, as given by Theorem 1.7. Then, for any observable x of M, the map $x:B\left(ℝ\right)\to \left[0,u\right]\subseteq G$ also defines a group-valued measure, i.e., a positive, additive, continuous map from B(R) to G. Conversely, every group-valued measure y:B(ℝ) → G such that y(ℝ)=u is an observable of M. The above construction yields the most general possible observable x in M. For instance, the observable of Example 1 arises as a group-valued measure x:B(ℝ) → G, where G is the ℓ-group of all measurable functions on Ω with the constant function 1 as the strong unit. To construct the observable of Examples 2 and 3, one can similarly let (G,u)=(ℝ, 1)

Having thus described the most general construction of an observable, let us now give a down-to-earth property of observables, when combined with states:

PROPOSITION 2.5

Let M be a σ-complete MV-algebra with an n-dimensional observable $x:ℬ\left({ℝ}^n\right)\to M$ and a state m: M→[0,1]. Define the composite map $m_x:ℬ\left({ℝ}^n\right)\to \left[0,1\right]$ by the stipulation that, for all $X\in ℬ\left({ℝ}^n\right)$ .

(6) $m_x\left(X\right)=\left(mоx\right)\left(X\right)=m\left(x\left(X\right)\right).$

Then m_x is a probability measure on B(ℝⁿ).

The proof is immediate. The above map m_x has the same role as the probability distribution of a random variable in the classical Kolmogorov theory. One can for instance investigate conditions ensuring the existence of the moments of observables in probability MV-algebras, in the following sense:

DEFINITION 2.7

Let (M, m) be a probability MV-algebra. Let x:M→ B(ℝ) be an observable of M. Then x is said to be integrable in (M, m), and we write $x\in L_m^1,$ if the expectation ⁷

exists. We say that x is square integrable, in symbols, $x\in L_m^2,$ if the dispersion (variance)

(8) ${\sigma }^2\left(x\right)={\displaystyle {\int }_{ℝ}{\left(t-E\left(x\right)\right)}^2}d{m}_x\left(t\right)={\displaystyle {\int }_{ℝ}{t}^{2}d{m}_x\left(t\right)-E{\left(x\right)}^2}$

exists.

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