On the observational equivalence of continuous-time deterministic and indeterministic descriptions

Abstract

This paper presents and philosophically assesses three types of results on the observational equivalence of continuous-time measure-theoretic deterministic and indeterministic descriptions. The first results establish observational equivalence to abstract mathematical descriptions. The second results are stronger because they show observational equivalence between deterministic and indeterministic descriptions found in science. Here I also discuss Kolmogorov’s contribution. For the third results I introduce two new meanings of ‘observational equivalence at every observation level’. Then I show the even stronger result of observational equivalence at every (and not just some) observation level between deterministic and indeterministic descriptions found in science. These results imply the following. Suppose one wants to find out whether a phenomenon is best modeled as deterministic or indeterministic. Then one cannot appeal to differences in the probability distributions of deterministic and indeterministic descriptions found in science to argue that one of the descriptions is preferable because there is no such difference. Finally, I criticise the extant claims of philosophers and mathematicians on observational equivalence.

Appendix

1.1 A.1. Semi-Markov and n-step Semi-Markov descriptions

First, we need the following definition.

Definition 9

A discrete stochastic description {Z _t ; t ∈ ℤ} is a family of random variables Z _t , t ∈ ℤ, from a probability space (Ω, Σ_Ω, ν) to a measurable space $(\bar{M},\Sigma_{\bar{M}})$.

Semi-Markov and n-step semi-Markov descriptions are defined via discrete-time Markov and n-step Markov descriptions (for a definition of these well-known descriptions, see Doob (1953), where it is also explained what it means for these descriptions to be irreducible and aperiodic).

A semi-Markov description²³ is defined with help of a discrete stochastic description {(S _k , T _k ), k ∈ ℤ}. {S _k ; k ∈ ℤ} describes the successive outcomes s _i visited by the process, where at time 0 the outcome is denoted by S ₀. T ₀ describes the time interval after which there is the first jump of the process after the time 0, T _− 1 describes the time interval after which there is the last jump of the process before time 0, and all other T _k similarly describe the time-intervals between jumps of the process. Because at time 0 the process takes the value S ₀ and the process is in S ₀ for the time u(S ₀), T _− 1 = u(S ₀) − T ₀.

Technically, {(S _k , T _k ), k ∈ ℤ} satisfies the following conditions: (i) S _k  ∈ S = {s ₁, ..., s _N }, N ∈ ℕ; $T_{k}\in U=\{u_{1},\ldots,u_{\bar{N}}\}$, $\bar{N}\in\mathbb{N}$, $\bar{N}\leq N$, for k ≠ 0, − 1, where $u_{i}\in\mathbb{R}^{+}$, $1\leq i \leq \bar{N}$; T ₀ ∈ (0, u(S ₀)], T _− 1 ∈ [0, u(S ₀)), where u : S → U, s _i → u(s _i ), is surjective; and hence $\bar{M}=S\times [0, \max_{i}u_{i}]$; (ii) $\Sigma_{\bar{M}}=\mathbb{P}(S)\times L([0, \max_{i}u_{i}])$, where L([0, max _i u _i ]) is the Lebesgue σ-algebra on [0, max _i u _i ]; (iii) {S _k ; k ∈ ℤ} is a stationary irreducible and aperiodic discrete-time Markov description with outcome space S; $p_{s_{i}}=P\{S_{0}=s_{i}\}>0$, for all i, 1 ≤ i ≤ N; (iv) T _k  = u(S _k ) for k ≥ 1, T _k  = u(S _k − 1) for k ≤ − 2, and T _− 1 = u(S ₀) − T ₀; (v) for all i, 1 ≤ i ≤ N, $P(T_{0}\in A\,|\,S_{0}=s_{i})=\int_{A} 1/u(s_{i})d\lambda$ for all A ∈ L((0, u(s _i )]), where L((0, u(s _i )]) is the Lebesgue σ-algebra and λ is the Lebesgue measure on (0, u(s _i )].

Definition 10

A semi-Markov description is a stochastic description {Z _t ; t ∈ ℝ} with outcome space S constructed via a stochastic description {(S _k , T _k ), k ∈ ℤ} as follows:

$$ \begin{array}{rll} Z_{t}&=&S_{0}\,\,\textnormal{for}\,\,-T_{-1}\leq t<T_{0}, \\ Z_{t}&=&S_{k}\,\,\textnormal{for}\,\,T_{0}+\ldots + T_{k-1}\leq t < T_{0}+\ldots + T_{k}; k\geq 1\,\,\textnormal{and thus}\,\,t\geq T_{0}, \\ Z_{t}&\!=\!&S_{-k}\,\textnormal{for}\!-\!T_{-1}\!-\!\ldots\!-\!T_{-k-1}\leq t < \!-\!T_{\!-1}\!-\!\ldots \!-\!T_{\!-k}; k\!\geq\! 1\,\textnormal{and thus}\,t<\!-\!T_{\!-\!1}, \end{array} $$

and for all i, 1 ≤ i ≤ N,

$$ P(Z_{0}=s_{i})=\frac{p_{s_{i}}u(s_{i})}{p_{s_{1}}u(s_{1})+\ldots +p_{s_{N}}u(s_{N})}. $$

(2)

Semi-Markov descriptions are stationary (Ornstein 1970 and Ornstein 1974, 56–61). In this paper I assume that the elements of U are irrationally related (u _i and u _j are irrationally related iff $\frac{u_{i}}{u_{j}}$ is irrational; and the elements of $U=\{u_{1},\ldots,u_{\bar{N}}\}$ are irrationally related iff for all i, j, i ≠ j, u _i and u _j are irrationally related).

n-step semi-Markov descriptions²⁴ generalise semi-Markov descriptions.

Definition 11

n-step semi-Markov descriptions are defined like semi-Markov descriptions except that condition (iii) is replaced by: (iii’) {S _k ; k ∈ ℤ} is a stationary irreducible and aperiodic discrete-time n-step Markov description with outcome space S and $p_{s_{i}}=P\{S_{0}=s_{i}\}>0$, for all i, 1 ≤ i ≤ N.

Again, n-step semi-Markov descriptions are stationary (Park 1982), and I assume that the elements of U are irrationally related.

1.2 A.2 Proof of Theorem 1

Iff for a measure-preserving deterministic description (M, Σ _M , μ, T _t ) there does not exist a n ∈ ℝ⁺ and a C ∈ Σ _M , 0 < μ(C) < 1, such that, except for a set of measure zero (esmz.) T _n (C) = C, then the following holds: for every nontrivial finite-valued observation function Φ : M → M _O , every k ∈ ℝ⁺ and {Z _t ; t ∈ ℝ} = {Φ(T _t ); t ∈ ℝ} there are o _i , o _j  ∈ M _O with 0 < P{Z_t + k = o _j | Z _t  = o _i } < 1.

Proof

We need the following definitions.□

Definition 12

A discrete measure-preserving deterministic description is a quadruple (M, Σ _M , μ, T) where (M, Σ _M , μ) is a probability space and T : M → M is a bijective measurable function such that T ^− 1 is measurable and μ(T(A)) = μ(A) for all A ∈ Σ _M .

Definition 13

A discrete measure-preserving deterministic description (M, Σ _M , μ, T) is ergodic iff there is no A ∈ Σ _M , 0 < μ(A) < 1, such that, esmz., T(A) = A.

(M, Σ _M , μ, T) is ergodic iff for all A, B ∈ Σ _M (Cornfeld et al. 1982, 14–15):

$$ \lim\limits_{n\rightarrow \infty}\frac{1}{n}\sum\limits_{i=1}^{n}(\mu(T^{n}(A)\cap B)-\mu(A)\mu(B))=0. $$

(3)

Definition 14

α = {α₁, ..., α _n }, n ∈ ℕ, is a partition of (M, Σ _M , μ) iff α _i  ∈ Σ _M , μ(α _i ) > 0, for all i, 1 ≤ i ≤ n, α _i  ∩ α _j  = ∅ for all i ≠ j, 1 ≤ i, j ≤ n, and $M=\bigcup_{i=1}^{n}\alpha_{i}$.

A partition is nontrivial iff n ≥ 2. Given two partitions α = {α ₁, ..., α _n } and β = {β ₁, ..., β _l } of (M, Σ _M , μ), α ∨ β is the partition {α _i  ∩ β _j | i = 1, ..., n;j = 1, ..., l}. Given a deterministic description (M, Σ _M , μ, T _t ), if α is a partition, T _t α = {T _t (α ₁), ..., T _t (α _n )}, t ∈ ℝ, is also a partition.

Note that any finite-valued observation function Φ (cf. Subsection 2.1) can be written as: $\Phi(m)=\sum_{i=1}^{n}o_{i}\chi_{\alpha_{i}}(m)$, M _O  = {o _i | 1 ≤ i ≤ n}, for some partition α of (M, Σ _M , μ), where χ _A is the characteristic function of A (cf. Cornfeld et al. 1982, 179).²⁵ It suffices to prove the following:

(*) Iff for (M, Σ _M , μ, T _t ) there does not exist an n ∈ ℝ⁺ and a C ∈ Σ _M , 0 < μ(C) < 1, such that, esmz., T _n (C) = C, then the following holds: for any nontrivial partition α = {α ₁, ..., α _r }, r ∈ ℕ, and all k ∈ ℝ⁺ there is an i ∈ {1, ..., r} such that for all j, 1 ≤ j ≤ r, μ(T _k (α _i ) ∖ α _j ) > 0.

For finite-valued observation functions are of the form $\sum_{l=1}^{r}o_{l}\chi_{\alpha_{l}}(m)$, where α = {α ₁, ..., α _r } is a partition and $M_{O}=\cup_{l=1}^{r}o_{l}$. Consequently, the right hand side of (*) expresses that for any nontrivial finite-valued Φ : M → M _O and all k ∈ ℝ⁺ there is an o _i  ∈ M _O such that for all o _j  ∈ M _O , P{Z _t + k = o _j | Z _t  = o _i } < 1, or equivalently, that there are o _i , o _j  ∈ M _O with 0 < P{Z _t + k = o _j | Z _t  = o _i } < 1.

• ⇐: Assume that there is an n ∈ ℝ⁺ and a C ∈ Σ _M , 0 < μ(C) < 1, such that, esmz., T _n (C) = C. Then for α = {C,M ∖ C} it holds that μ(T _n (C) ∖ C) = 0 and μ(T _n (M ∖ C) ∖ (M ∖ C)) = 0.

• ⇒: Assume that the conclusion of (*) does not hold, and hence that there is a nontrivial partition α and a k ∈ ℝ⁺ such that for each α _i there is an α _j with, esmz., $T_{k}(\alpha_{i})\subseteq \alpha_{j}$. From the assumptions it follows that for every k ∈ ℝ⁺ the discrete deterministic description (M, Σ _M , μ, T _k ) is ergodic (cf. Definition 13).

  • Case 1: For every i there is a j such that, esmz., T _k (α _i ) = α _j . Because the discrete description (M, Σ _M , μ, T _k ) is ergodic (Eq. 3), there is an h ∈ ℕ such that, esmz., T _kh (α₁) = α₁. But this is in contradiction with the assumption that it is not the case that there exists an n ∈ ℝ⁺ and a C ∈ Σ _M , 0 < μ(C) < 1, such that, esmz., T _n (C) = C.

  • Case 2: There exists an i and a j with, esmz., T _k (α _i ) ⊂ α _j and with μ(α _i ) < μ(α _j ). Because the discrete description (M, Σ _M , μ, T _k ) is ergodic (Eq. 3), there is a h ∈ ℕ such that, esmz., $T_{hk}(\alpha_{j})\subseteq\alpha_{i}$. Hence μ(α _j ) ≤ μ(α _i ), contradicting μ(α _i ) < μ(α _j ) ≤ μ(α _i ).

1.3 A.3 Proof of Theorem 4

Theorem 4

Let (M, Σ _M , μ, T _t ) be a continuous Bernoulli system. Then for every finite-valued Φ and every ε > 0 a semi-Markov description {Z _t , t ∈ ℝ} weakly (Φ, ε)-simulates (M, Σ _M , μ, T _t ).

Proof

Let (M, Σ _M , μ, T _t ) be a continuous Bernoulli system, let Φ : M → S, S = {s ₁, ..., s _N }, N ∈ ℕ, be a surjective observation function, and let ε > 0. Theorem 3 implies that there is a surjective observation function $\Theta:M\rightarrow S,\,\,\Theta(m)=\sum_{i=1}^{N}s_{i}\chi_{\alpha_{i}}(m)$, for a partition α (cf. Definition 14), such that {Y _t  = Θ(T _t ); t ∈ ℝ} is an n-step semi-Markov description with outcomes s _i and corresponding times u(s _i ) which strongly (Φ, ε)-simulates (M, Σ _M , μ, T _t ). □

I need the following definition:

Definition 15

$(M_{2},\Sigma_{M_{2}},\mu_{2},T^{2}_{t})$ is a factor of $(M_{1},\Sigma_{M_{1}},\mu_{1},T^{1}_{t})$ (where both deterministic descriptions measure-preserving) iff there are $\hat{M}_{i}\subseteq M_{i}$ with $\mu_{i}(M_{i}\setminus\hat{M}_{i})=0$, $T^{i}_{t}\hat{M}_{i}\subseteq\hat{M}_{i}$ for all t (i = 1, 2), and there is a function $\phi:\hat{M}_{1}\!\rightarrow \!\hat{M}_{2}$ such that (i) $\phi^{-1}(B)\!\in\!\Sigma_{M_{1}}$ for all $B\!\in\!\Sigma_{M_{2}},A\subseteq\hat{M}_{2}$; (ii) $\mu_{1}(\phi^{-1}(B))=\mu_{2}(B)$ for all $B\in\Sigma_{M_{2}},\,B\subseteq\hat{M}_{2}$; (iii) $\phi(T^{1}_{t}(m))=T^{2}_{t}(\phi(m))$ for all $m\in\hat{M}_{1},$t ∈ ℝ.

Note that the deterministic representation (X, Σ _X , μ _X , W _t , Λ₀) of {Y _t ; t ∈ ℝ} is a factor of (M, Σ _M , μ, T _t ) via ϕ(m) = r _m (r _m is the realisation of m) (cf. Ornstein and Weiss 1991, 18).

Now I construct a measure-preserving deterministic description (K, Σ _K , μ _K , R _t ) as follows. Let (Ω, Σ_Ω, μ _Ω, V, Ξ₀), $\Xi_{0}(\omega)=\sum_{i=1}^{N}s_{i}\chi_{\beta_{i}}(\omega)$, where β is a partition, be the deterministic representation of {S _k ; k ∈ ℤ}, the irreducible and aperiodic n-step Markov description corresponding to {Y _t ; t ∈ ℝ}. Let f : Ω → {u ₁, ..., u _N }, f(ω) = u(Ξ₀(ω)). Define K as $\cup_{i=1}^{N}K_{i}=\cup_{i=1}^{N}(\beta_{i}\times [0,u(s_{i})))$. Let $\Sigma_{K_{i}}$, 1 ≤ i ≤ N, be the product σ-algebra (Σ_Ω ∩ β _i ) × L([0, u(s _i ))) where L([0, u(s _i ))) is the Lebesgue σ-algebra of [0, u(s _i )). Let $\mu_{K_{i}}$ be the product measure

$$ \left(\mu_{\Omega}^{\Sigma_{\Omega}\cap\beta_{i}}\times\lambda([0,u(s_{i})))\right)/\sum\limits_{j=1}^{N}u\left(s_{j}\right)\mu_{\Omega}(\beta_{j}), $$

(4)

where λ([0, u(s _i ))) is the Lebesgue measure on [0, u(s _i )) and $\mu_{\Omega}^{\Sigma_{\Omega}\cap\beta_{i}}$ is the measure μ _Ω restricted to Σ_Ω ∩ β _i . Let Σ _K be the σ-algebra generated by $\cup_{i=1}^{N}\Sigma_{K_{i}}$. Define a pre-measure $\bar{\mu}_{K}$ on $H=(\cup_{i=1}^{N}(\Sigma_{\Omega}\cap\beta_{i}\times L([0,s_{i}))))\cup K$ by $\bar{\mu}_{K}(K)=1$ and $\bar{\mu}_{K}(A)=\mu_{K_{i}}(A)$ for $A\in\Sigma_{K_{i}}$, and let μ _K be the unique extension of this pre-measure to a measure on Σ _K . Finally, R _t is defined as follows: let the state at time zero be represented by (k, v) ∈ K, k ∈ Ω, v < f(k); the representation of the state moves vertically with unit velocity, and just before it reaches (k, f(k)) it jumps to (V(k), 0) at time f(k) − v; then it again moves vertically with unit velocity, and just before it reaches (V(k), f(V(k)))) it jumps to (V ²(k), 0) at time f(V(k)) + f(k) − v, and so on. (K, Σ _K , μ _K , R _t ) is a measure-preserving deterministic description (a ‘flow built under the function f’). (X, Σ _X , μ _X , W _t ) is proven to be isomorphic (via a function ψ) to (K, Σ _K , μ _K , R _t ) (Park 1982).

Consider $\gamma\!=\!\{\gamma_{1},\!\ldots,\! \gamma_{l}\}\!=\!\beta\vee V\beta\vee\ldots\vee V^{n-1}\beta$ and $\Pi(\omega)=\sum_{j=1}^{l}o_{j}\chi_{\gamma_{j}}(\omega)$, o _i  ≠ o _j for i ≠ j, 1 ≤ i, j ≤ l. I now show that $\{B_{t}=\Pi(V^{t}(\omega));\,\,t\in\mathbb{Z}\}$ is an irreducible and aperiodic Markov discrete-time stochastic description. By construction, for all t and all i, 1 ≤ i ≤ l, there are q _i,0, ..., q _{i,n − 1} ∈ S such that

$$ P\{B_{t}=o_{i}\}=P\left\{S_{t}=q_{i,0}, S_{t+1}=q_{i,1},\ldots,S_{t+n-1}=q_{i,n-1}\right\}. $$

(5)

Therefore, for all k ∈ ℕ and all i, j ₁, ..., j _k , 1 ≤ i, j ₁, ..., j _k  ≤ l:

$$ \begin{array}{rll} P\left\{B_{t+1}\right. &=& \left.o_{i}\,|\,B_{t}=o_{j_{1}},\ldots,B_{t-k+1}=o_{j_{k}}\right\} \\ &=& P \left\{S_{t+1} = q_{i,0},...,S_{t+n} = q_{i,n-1}|S_{t}= q_{j_{1},0},...,S_{t+n-1}\right. \\ &&\qquad\qquad =\left. q_{i,n-2}, S_{t-1} = q_{j_{2},0},...,S_{t-k+1} = q_{j_{k},0}\right\} \\ &=&P\left\{S_{t+1}=q_{i,0},\ldots,S_{t+n}=q_{i,n-1}\,|\,S_{t}=q_{j_{1},0},\ldots,S_{t+n-1}=q_{i,n-2}\right\} \\ &=&P\left\{B_{t+1}=o_{i}\,|\,B_{t}=o_{j_{1}}\right\}, \end{array} $$

(6)

if $P\{B_{t+1}=o_{i},B_{t}=o_{j_{1}},\ldots,B_{t-k+1}=o_{j_{k}}\}>0$. Hence {B _t ; t ∈ ℤ} is a Markov description. Now a discrete measure-preserving deterministic description (M, Σ _M , μ, T) is mixing iff for all A, B ∈ Σ _M

$$ \lim\limits_{t\rightarrow\infty}\mu(T^{t}(A)\cap B)=\mu(A)\mu(B). $$

(7)

The deterministic representation of every irreducible and aperiodic n-step Markov description, and hence (Ω, Σ_Ω, μ _Ω, V, Ξ₀), is mixing (Ornstein 1974, 45–47). This implies that {B _t ; t ∈ ℤ} is irreducible and aperiodic.

Let $\Delta(k)=\sum_{i=1}^{l}o_{i}\chi_{\gamma_{i}\times[0,u(o_{i}))}(k)$, where u(o _i ), 1 ≤ i ≤ l, is defined as follows: u(o _i ) = u(s _r ) where $\gamma_{i}\subseteq \beta_{r}$. It follows immediately that {X _t  = Δ(R _t ); t ∈ ℝ} is a semi-Markov description. Consider the surjective function Ψ : M → {o ₁, ..., o _l }, Ψ (m) = Δ(ψ(ϕ(m))) for $m\in\hat{M}$ and o ₁ otherwise. Recall that (X, Σ _X , μ _X , W _t ) is a factor (via ϕ) of (M, Σ _M , μ, T _t ) and that (X, Σ _X , μ _X , W _t ) is isomorphic (via ψ) to (K, Σ _K , μ _K , R _t ). Therefore, {Z _t  = Ψ(T _t ); t ∈ ℝ} is a semi-Markov process with outcomes o _i and times u(o _i ), 1 ≤ i ≤ l.

Now consider the surjective observation function Γ : {o ₁, ..., o _l } → S, where Γ(o _i ) = s _r for $\gamma_{i}\subseteq\beta_{r}$, 1 ≤ i ≤ l. By construction, esmz., Γ(Ψ(T _t (m))) =Θ(T _t (m)) = Y _t (m) for all t ∈ ℝ. Hence, because {Y _t ; t ∈ ℝ} strongly (Φ, ε)-simulates (M,Σ _M , μ, T _t ), μ({m ∈ M | Γ(Ψ(m)) ≠ Φ(m)}) < ε.

1.4 A.4 Proof of Proposition 1

Proposition 1

Let (M, Σ _M , μ, T _t ) be a measure-preserving deterministic description where (M, d _M ) is separable and where Σ _M contains all open sets of (M, d _M ). Assume that (M, Σ _M , μ, T _t ) satisfies the assumption of Theorem 1 and has finite KS-entropy. Then for every ε > 0 there is a stochastic description {Z _t ; t ∈ ℝ} with outcome space $M_{O}=\cup_{l=1}^{h}o_{l}$, h ∈ ℕ, such that {Z _t ; t ∈ ℝ} is ε-congruent to (M, Σ _M , μ, T _t ), and for all k ∈ ℝ⁺ there are o _i , o _j  ∈ M _O such that 0 < P{Z_t + k = o _j | Z _t  = o _i } < 1.

Proof

A partition α (cf. Definition 14) is generating for the discrete measure-preserving description (M, Σ _M , μ, T) iff for every A ∈ Σ _M there is an n ∈ ℕ and a set C of unions of elements in $\vee_{j=-n}^{n}T^{j}(\alpha)$ such that μ((A ∖ C) ∪ (C ∖ A)) < ε (Petersen 1983, 244). α is generating for the measure-preserving description (M, Σ _M , μ, T _t ) iff for all A ∈ Σ _M there is a τ ∈ ℝ⁺ and a set C of unions of elements in $\bigcup_{\textnormal{all}\,\,m}\bigcap_{t=-\tau}^{\tau}(T^{-t}(\alpha(T^{t}(m))))$ such that μ((A ∖ C) ∪ (C ∖ A)) < ε (α(m) is the set α _j  ∈ α with m ∈ α _j ). □

By assumption, there is a $t_{0}\in\mathbb{R}^{+}$ such that the discrete description $(M,\Sigma_{M},\mu,T_{t_{0}})$ is ergodic (cf. Definition 13). For discrete ergodic descriptions Krieger’s (1970) theorem implies that there is a partition α which is generating for $(M,\Sigma_{M},\mu,T_{t_{0}})$ and hence generating for (M, Σ _M , μ, T _t ). Since (M, d _M ) is separable, for every ε > 0 there is a r ∈ ℕ and m _i  ∈ M, 1 ≤ i ≤ r, such that $\mu(M\setminus\cup_{i=1}^{r}B(m_{i},\frac{\varepsilon}{2}))<\frac{\varepsilon}{2}$. Because α is generating for $(M,\Sigma_{M},\mu,T_{t_{0}})$, for each $B(m_{i},\frac{\varepsilon}{2})$ there is an n _i  ∈ ℕ and a C _i of union of elements in $\vee_{j=-n_{i}}^{n_{i}}T_{jt_{0}}(\alpha)$ such that $\mu((B(m_{i},\frac{\varepsilon}{2})\setminus C_{i})\cup (C_{i}\setminus B(m_{i},\frac{\varepsilon}{2})))<\frac{\varepsilon}{2r}$. Let $n\!=\!\max\{n_{i}\},\,\,\beta\!=\!\{\beta_{1},\ldots,\beta_{l}\}\!=\!\vee_{j=-n}^{n}T_{jt_{0}}(\alpha)$ and $\Psi\!(m)=\!\sum_{i=1}^{l}o_{i}\chi_{\beta_{i}}(m)$ with o _i  ∈ β _i . Since Ψ is a finite-valued, Theorem 1 implies that for the stochastic description {Ψ(T _t ); t ∈ ℝ} for all k ∈ ℝ⁺ there are o _i , o _j such that 0 < P{Z _t + k = o _j | Z _t  = o _i } < 1. Because α is generating for (M, Σ _M , μ, T _t ), β is generating too. This implies that (M, Σ _M , μ, T _t ) is isomorphic (via a function ϕ) to the deterministic representation $(M_{2},\Sigma_{M_{2}},\mu_{2},T^{2}_{t},\Phi_{0})$ of {Z _t ; t ∈ ℝ} (Petersen 1983, 274). And, by construction, d _M (m, Φ₀(ϕ(m))) < ε except for a set in M smaller than ε.