INFINITELY FINE PARTITIONS OF MEASURE SPACES

VLADIMIR G. TROITSKY

1. Infinitely fine partitions

Throughout this paper (X , A) always stands for a measurable space, i.e., X is an arbitrary set and A is an algebra of its subsets.

By ^{∗}X we denote the nonstandard extension of X. We assume that the nonstandard model is κ-saturated where κ is the cardinality of A.

1.1. Definition. A measurable partition P of ^{∗}X is said to be an infinitely fine partition (abbreviated ifp) if ^{∗}A = ⋃{p ∈ P | p ⊆ ^{∗}A} for every measurable set A, or, equivalently, if P is finer than every standard partition.

It follows from the saturation principle that every measurable space (and even a Boolean algebra) has a hyperfinite ifp. From now on the symbol P stands for a fixed hyperfinite ifp of the measurable space (X , A). Infinitely fine partitions were originally introduced in [Loe72]. Some properties of ifp’s were investigated in [Tro93].

Notice that if A contains all the singletons, then every standard singleton belongs to P.

1.2. Definition. We say that p_{1}, p_{2} ∈ P are equivalent (denoted p_{1} ∼ p_{2}) if P \ {p_{1}, p_{2}} ∪ {p_{1} ∪ p_{2}} is again an ifp.

1.3. Theorem. An element of P is not equivalent to any other element of P if and only if it is standard.

Proof. Obviously every standard elements of P is not equivalent to another element of P. Suppose that p ∈ P is non-standard. Consider

Vladikavkaz Mathematical Journal, 1(1999), no. 3, 53–59.

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the collection q = {A ∈ A | p ⊆ ^{∗}A}. Let S be a finite subset of q and put S = ⋂ S, then S ∈ A. Evidently p ⊆ ^{∗}S, but since p is nonstandard, it follows that p ≠ ^{∗}S. Therefore, there exists p′ ∈ P such that p′ ⊂ ^{∗}S and p′ ≠ p. Since S was chosen as an arbitrary finite sub-collection of q, by the saturation principle there exists an element p_{1} ∈ P distinct from p contained in the extension of every set belonging to q, so that p ∼ p_{1}. □

It was shown in [Loe72] that a standard bounded measurable function is approximately constant on every element of an ifp.

1.4. Lemma. Let p_{1}, p_{2} ∈ P, then p_{1} ∼ p_{2} if and only if every standard bounded measurable function f: X → ℝ takes approximately equal values on p_{1} and p_{2}.

Proof. The implication from left to right follows from the fact that P \ {p_{1}, p_{2}} ∪ {p_{1} ∪ p_{2}} is an ifp. The converse can be easily obtained by considering characteristic functions of standard measurable sets. □

2. Nonatomicity

In this section we present some interpretations of nonatomicity in terms of infinitely fine partitions.

Suppose now that (X , A) is endowed with a standard finitely additive measure μ. We introduce the following notations:

\begin{aligned}P_{+} & = & \{p ∈ P | ^{∗}μ(p) > 0\} & P_{0} & = & \{p ∈ P | ^{∗}μ(p) = 0\} \\ A_{+} & = & \{A ∈ A | μ(A) > 0\} & A_{0} & = & \{A ∈ A | μ(A) = 0\} \end{aligned}

The elements of P_{+} will be referred to as the essential elements of P. Notice that every points of an essential element of P is random (recall that a point x ∈ ^{∗}X is said to be random if x ∉ ^{∗}N for each N ∈ A_{0}).

2.1. Definition. Let p be an essential element of P. We say that p is essentially joinable if p is equivalent to another essential element of P. We call p essentially divisible if it can be written as a union of two disjoint sets of positive measure.

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Recall that a set F ∈ A_{+} is said to be an atom of μ if for every measurable set E ⊆ F we have either μ(E) = 0 or μ(E) = μ(F). A measure is said to be nonatomic if it has no atoms. A measure is said to be strongly continuous if for every ε > 0 there exists a finite measurable partition of X into sets of measure less than ε. A measure μ is said to be strongly nonatomic if for every set F ∈ A_{+} and every 0 < c < μ(F) there exists a measurable set E ⊂ F such that μ(E) = c.

It was shown in [BRBR83] that strong nonatomicity implies strong continuity, strong continuity implies nonatomicity, and all the three properties are equivalent for sigma-additive measures.

2.2. Theorem. The following statements are equivalent:

(1) every essential element of P is essentially divisible;

(2) every essential element of P is essentially joinable;

(3) μ is nonatomic.

Proof. To prove the implication (1)⇒(2), consider an essential element p ∈ P. By the assumption, it is essentially divisible. Let q = {A ∈ A | p ⊂ ^{∗}A}, and consider a finite sub-collection S of q. Let S = ⋂ S. Evidently, S is a standard set and p ⊂ ^{∗}S. Essential divisibility of p implies that there exists p_{1} ∈ ^{∗}A_{+} such that 0 < ^{∗}μ(p_{1}) < ^{∗}μ(p) ⩽ μ(S). By the transfer principle there exists a standard set B ⊂ S such that the both B and S \ B are of positive measure. Assume without loss of generality that p_{1} ⊂ ^{∗}B. Obviously there is an essential element p_{2} ⊂ ∗S \^{∗}B ⊂ ^{∗}S. Since S was chosen arbitrarily, it follows by saturation that there exists p′ ∈ P_{+} distinct from p, such that p′ ⊆ ^{∗}A for each A ∈ q, and it follows that p′ ∼ p.

To prove (2)⇒(3) take an arbitrary set A ∈ A_{+}. There exists p ∈ P_{+} such that p ⊆ ^{∗}A. Since p is essentially joinable, then there is p′ ∈ P_{+} such that p′ ⊂ ^{∗}A \ p and it follows that ^{∗}μ(p) < μ(A). By using the transfer principle we obtain the conclusion.

The implication (3)⇒(1) is straightforward. □

2.3. Theorem. A measure is strongly continuous if and only if the measure of every element of P is infinitesimal.

Proof. The proof is elementary. □

2.4. Theorem. A measure μ is strongly nonatomic if and only if (∀p ∈ P_{+})(∀λ ∈ ^{∗}[0, 1])(∃p′ ∈ ^{∗}A_{+})(p′ ⊂ p & \frac{^{∗}μ(p′)}{^{∗}μ(p)} = λ).

Proof. The implication from left to right is obvious. To prove the converse implication, fix F ∈ A_{+} and c ∈ (0, μ(F)). Consider an internal measurable subset E of ^{∗}F such that

(1) E = ⋃{p ∈ P | p ⊂ E};

(2) ^{∗}μ(E) ⩽ c;

(3) p ⊈ E implies ^{∗}μ(E ∪ p) > c for each p ∈ P_{+}. Such a set exists because ^{∗}F is a union of a hyperfinite number of elements of P. We claim that ^{∗}μ(E) = c. Indeed, if ^{∗}μ(E) < c, then we could take any p ⊆ ^{∗}F \ E and split it into the disjoint union of p_{1} and p_{2} so that ^{∗}μ(E ∪ p_{1}) = c, which would contradict to the assumptions on E. The transfer principle completes the proof. □

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3. Representation of L_{∞}

In the following section we extend a results obtained by P. Loeb in [Loe72]. For this subsection, let μ be a standard sigma-additive measure, P a hyperfinite ifp for a standard measurable space (X , A), and L_{∞} stands for L_{∞}(X , A, μ). In [Loe72] Loeb introduced a map T_{0} : L_{∞} → ℝ^{P} by the following rule. For every p ∈ P_{+} pick a point c_{p} ∈ p. For f ∈ L_{∞} define

(T_{0}(f))_{p} = \{ \begin{matrix}^{∘}f(c_{p}) & if p ∈ P_{+}; \\ 0 & if p ∈ P_{0}. \end{matrix}

Loeb showed in [Loe72] that the variation of a standard bounded measurable function on each element of an ifp is infinitesimal, so that T_{0} does not depend on the choise of c_{p}’s. Loeb proved that a vector

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v ∈ ℝ^{P} equals T_{0}(f) for some f ∈ L_{∞} if and only if the following three conditions hold:

(1) v_{p} = 0 for all p ∈ P_{0};

(2) max_{p∈P}|v_{p}| is nearstandard;

(3) for each p in P_{+} and for each ε ∈ ℝ_{+} there exists a set A ∈ A such that p ⊆ ^{∗}A and |v_{p} − v_{p′}| < ε for every essential p′ ⊆ ^{∗}A.

3.1. Lemma. Condition (3) above is equivalent to the following statement:

(3’) p_{1} ∼ p_{2} implies v_{p_{1}} ≈ v_{p_{2}} for any p_{1}, p_{2} ∈ P_{+}.

Proof. Suppose that p_{1} ∼ p_{2} for p_{1}, p_{2} ∈ P_{+}. Then it follows from (3) that |v_{p_{1}} − v_{p_{2}}| < ε for every ε ∈ ℝ_{+}, hence v_{p_{1}} ≈ v_{p_{2}}, so that (3) implies (3’). To show the reverse implication, assume that there exists p ∈ P_{+} and ε ∈ ℝ_{+} such that for each A ∈ A satisfying p ⊆ ^{∗}A one can find p′ ∈ P_{+} such that p′ ⊆ ^{∗}A and |v_{p} − v_{p′}| > ε. Let q = {A ∈ A | p ⊆ ^{∗}A} and consider a finite sub-collection S of q. Let S = ⋂ S. By the assumption, there exists an essential p′ ⊆ ^{∗}S such that |v_{p} − v_{p′}| ⩾ ε. By the saturation principle there exists an element p′ ∈ P_{+}, such that p′ ⊆ ^{∗}A and |v_{p} − v_{p′}| ⩾ ε, but this contradicts to (3’). □

4. Monads of ifp

Again, let P be an ifp for a standard measurable space (X , A). For p ∈ P we denote by [p] the equivalence class of p, i.e., [p] = {p′ ∈ P | p′ ∼ p}. The union of the elements of [p] will be denoted by m_{p} and referred to as a monad of P. Since “∼” is an equivalence relation, 𝔐 = {m_{p}}_{p∈P} is an partition of ^{∗}X. It is easy to see that 𝔐 is exactly the partition of X generated by all standard measurable sets. It implies, in particular, that the monads are independent of a particular ifp.

Let Q be the Stone space of the algebra A. Recall that the points of Q are the ultra-filters on A. It is well known that there exists a

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canonical boolean algebra isomorphism ι between A and the algebra Clop Q of all clopen subsets of Q given by ι(A) = {q ∈ Q | A ∈ q}.

Notice that there is a one-to-one correspondences between the monads of P, the ultra-filters on A, the zero-one measures on A, and the points of Q. Consider a monad m of P. The corresponding ultrafilter q_{m} is {A ∈ A | m ⊆ ^{∗}A}, i.e., the elements of q_{m} are exactly the standard measurable sets containing the monad m. On the other hand, since m = ⋃[p] where [p] is exactly the equivalence class of those elements of P which are contained in each A in q_{m}, we have m = ⋂ q_{m}.

Any monad m corresponds to some zero-one measure on A defined by

δ_{m}(A) = \{ \begin{matrix}1 & if m ⊆ ^{∗}A, \\ 0 & otherwise. \end{matrix}

It is easy to see that q_{m} = {A ∈ A | δ_{m}(A) = 1}. Finally, for any monad m we can consider the ultra-filter q_{m} as a point of the Stone space Q of A. Notice that the set ⋃{ι(p) | p ∈ P, p ⊆ m} is the usual topological monad of q_{m} in Q. Notice also, that {q_{m} | m ⊂ ^{∗}A} = ι(A) for any A in A.

It follows from [Loe72, Proposition 4.2] that there is a one-to-one correspondence between the monads of P and the multiplicative linear functionals on L_{∞}(X). Namely, if m is a monad of P, then the map f ↦ ^{∘}(^{∗}f)_{|m} is the multiplicative linear functional corresponding to m.

Consider a monad m of P. Since q_{m} is a standard ultra-filter on A, it follows that ^{∗}q_{m} contains exactly one element of P. This element will be referred as the central element of m, we will denote it by p_{m}. Obviously p_{m} ⊆ m. We mentioned before that {ι(p) | p ∈ P, p ⊆ m} form a partition of the topological monad of the (standard) point q_{m} in Q. Then p_{m} can be characterized as the element of P for which q_{m} ∈ ^{∗}ι(p_{m}). Finally, central elements can also be characterized in terms of zero-one measures. Namely, p = p_{m} if and only if ^{∗}δ_{m}(p) = 1. Denote by P_{c} the collection of all central elements of P.

It will be shown that properties of measures are essentially determined by their values at the central elements. For the rest of this section we suppose that μ is a standard finitely additive measure on A. We assume for simplicity that μ is finite.

4.1. Theorem. If p is a central element with ^{∗}μ(p) standard, then (1) ^{∗}μ(p′) = 0 for any noncentral p′ equivalent to p;

(2) There exists a set A ∈ A such that p ⊆ ^{∗}A and μ(A) = ^{∗}μ(p).

Proof. It follows from p ∈ P_{c} that p = p_{m} for some monad m of P. Since p ∈ ^{∗}q_{m} and ^{∗}μ(p) is standard, then by the transfer principle we can find a standard set A ∈ q_{m} such that μ(A) = ^{∗}μ(p). This proves (2). Finally, (1) follows immediately from (2). □

4.2. Corollary. Suppose that p is a central element of measure zero. If p′ ∼ p then p′ is of measure zero.

4.3. Theorem. If p is a central element of P, then for every standard ε > 0 there exists a standard set A ∈ A such that m_{p} ⊆ ^{∗}A and μ(A) ⩽ ^{∗}μ(p_{m}) + ε.

Proof. Let λ = ^{∗}μ(p) + \frac{ε}{2}, then ^{∗}μ(p) < ^{∘}λ. Since p ∈ ^{∗}q_{m} then, by the transfer principle, there exists a standard set A ∈ q_{m} such that μ(A) < ^{∘}λ < ^{∗}μ(p_{m}) + ε. □

4.4. Corollary. The measure of every noncentral element is infinitesimal. Moreover, the union of any collection of non-central elements of m has infinitesimal measure. Furthermore, if m is a monad of P and D ⊂ m for some D ∈ ^{∗}A such that q_{m} ∉ ^{∗}ι(D), then ^{∗}μ(D) ≈ 0.

4.5. Remark. There is another important consequence of Theorem 4.3. We have noticed before that the monads do not depend on a particular ifp, but only on the algebra. Now it follows immediately from Theorem 4.3 that if m is a monad, then ^{∘}(^{∗}μ(p_{m})) does not depend on a particular ifp either.

4.6. Lemma. If p is a central element of P then ^{∗}μ(p) ⩾ ^{∘}(^{∗}μ(p)).

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Proof. For every standard A ∈ q_{m_{p}} we have p ⊂ A, and, therefore, ∘(^{∗}μ(p)) ⩽ μ(A). By the transfer principle it follows that ^{∘}(^{∗}μ(p)) ⩽ ∗μ(p). □

5. Sobczik-Hammer Decomposition Theorem

The technique of ifp gives us an opportunity to present a simple proof of the Sobczik-Hammer Decomposition Theorem.

5.1. Theorem (Sobczik-Hammer Decomposition Theorem). Let μ be a finite finitely additive measure on a measurable space (X , A). Then there exists a sequence (δ_{n})_{n∈ℕ} of distinct zero-one measures on A, a sequence (a_{n})_{n∈ℕ} of nonnegative real numbers, and a strongly continuous measure μ̄ on A, such that ∑_{n=1}^{∞} a_{n} < ∞ and μ = μ̄ + ∑_{n=1}^{∞} a_{n}δ_{n}. Further, this decomposition is unique up to the order of terms.

Proof. Let P be a hyperfinite ifp for (X , A). Let p_{1} be an element of P of maximum measure, put a_{1} = ^{∘}((^{∗}μ(p_{1})). For δ_{1} we take the zero-one measure corresponding to m_{p_{1}}. If μ is strongly continuous, then a_{1} = 0 by Theorem 2.3. Otherwise, it follows from Corollary 4.4 that p_{1} is central. Then Lemma 4.6 implies that a_{1} ⩽ ^{∗}μ(p_{1}). It follows that μ_{1} = μ − a_{1}δ_{1} is a standard measure which is nonnegative on each element of P, hence μ_{1} is nonnegative. We proceed in the similar fashion defining μ_{n+1} = μ_{n} − a_{n}δ_{n}, etc. This process may stop after a finite number of steps if a_{n} = 0 for some n. In this case we get μ = ∑_{i=1}^{n} a_{i}δ_{i} + μ_{n} where μ_{n} is a strongly continuous measure, so that the conclusion of the theorem is satisfied. Otherwise, we and obtain a decreasing sequence (a_{n})_{n∈ℕ} of standard non-negative reals, a sequence (δ_{n})_{n∈ℕ} of standard distinct zero-one measures, and a sequence (μ_{n})_{n∈ℕ} of standard non-negative measures, such that μ_{n} = μ − ∑_{i=1}^{n} a_{i}δ_{i} for every n ∈ ℕ. In particular, μ(X) ⩾ ∑_{i=1}^{n} a_{i}δ_{i}(X) = ∑_{i=1}^{n} a_{i} for every natural n ∈ ℕ; it follows that ∑_{i=1}^{∞} a_{i} < ∞.

It can be easily verified that μ̄ defined by μ̄ = μ − ∑_{n=1}^{∞} a_{n}δ_{n} is a standard nonnegative measure, and μ̄ ⩽ μ_{n} for all n ∈ ℕ. Assume that μ̄ is not strongly continuous, then there exists p ∈ P such

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that st(^{∗}μ̄(p)) > 0. Since lim_{n} a_{n} = 0, we can find n ∈ ℕ such that st(^{∗}μ̄(p)) > a_{n} ⩾ st(^{∗}μ_{n−1}(p)); but this contradicts to the fact that μ̄ ⩽ μ_{n} for each n ∈ ℕ.

We see that the numbers a_{n} and measures δ_{n} are completely determined by the values of μ on the central elements of P up to the order. Thus, the constructed decomposition is unique. □

The concept of an infinitely fine partition also gives us an opportunity to prove the Sobczik-Hammer Decomposition Theorem for vector measures. Let Y be a standard Banach space, and let F: A → Y be a standard Y -valued measure on (X , A). Recall that the variation of F is a real-valued measure given by |F |(A) = sup_{π} ∑_{B∈π}∥F (B)∥, where sup is taken over all finite measurable partitions π of A. A vector measure is said to be nonatomic (strongly continuous, strongly nonatomic) if the same is true for its variation.

To prove a vector analogue of Sobczik-Hammer Theorem we need the following lemma. As usually, we assume that P is a hyperfinite ifp for (X , A).

5.2. Lemma. If F is a standard vector measure of bounded variation on (X , A); then |^{∗}F |(p) ≈ ∥^{∗}F (p)∥ for each p ∈ P.

Proof. Clearly |F |(A) ⩾ ∥F (A)∥ for any measurable A. If p ∈ P \ P_{c} then |^{∗}F |(p) ≈ 0 by Corollary 4.4, so that |^{∗}F |(p) ≈ ∥^{∗}F (p)∥ holds trivially.

Suppose that p ∈ P_{c}. Fix an infinitesimal ε > 0 and consider a hyperfinite measurable partition π of p such that |^{∗}F |(p) ⩽ ∑_{p′∈π}∥^{∗}F (p′)∥+ε. Then P \ {p} ∪ π is again a hyperfinite ifp. Let p_{π} be the central element of this new ifp, corresponding to the monad of p. By Corollary 4.4 |^{∗}F |(p \ p_{π}) ≈ 0, so that

∥^{∗}F (p_{π})∥ ⩽ |^{∗}F |(p) ⩽ ∑_{p′∈π}∥^{∗}F (p′)∥+ε ⩽ ∑_{p′∈π,p′≠p_{π}}|^{∗}F |(p′)+∥^{∗}F (p_{π})∥+ε

= |^{∗}F |(p \ p_{π}) + ∥^{∗}F (p_{π})∥ + ε ≈ ∥^{∗}F (p_{π})∥.

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It follows that ∥^{∗}F (p_{π})∥ ≈ |^{∗}F |(p). On the other hand, it follows from ∗F (p) = ^{∗}F (p_{π}) +^{∗}F (p \ p_{π}) and ∥^{∗}F (p \ p_{π})∥ ⩽ |^{∗}F |(p \ p_{π}) ≈ 0 that ∥^{∗}F (p)∥ ≈ ∥^{∗}F (p_{π})∥, so that ∥^{∗}F (p)∥ ≈ |^{∗}F |(p). □

Now we can proceed with the vector-valued version of the Sobczik-Hammer Theorem. A similar result can be found in [KM89], but the proof presented there is based on completely different concepts.

5.3. Theorem (Sobczik-Hammer Decomposition Theorem for vector measure). Let F be a standard Y -valued measure of bounded variation on (X , A) such that the range of F is relatively compact. Then there exists a strongly additive vector measure F̄, a sequence (x_{n})_{n∈ℕ} ⊂ E, and a sequence (δ_{n})_{n∈ℕ} of distinct zero-one measures on (X , A), such that F = F̄ + ∑_{n=1}^{∞} x_{n}δ_{n}. Further, this decomposition is unique up to the order of terms.

Proof. The proof is analogous to the proof of Theorem 5.1. Consider a hyperfinite ifp P for (X , A). Let p_{1} be an element of P of maximum value of |^{∗}F |. Since F has compact range we are guaranteed that ∗F (p_{1}) is nearstandard. Let x_{1} = st(^{∗}F (p_{1})), let δ_{1} be the zero-one measure corresponding to [p], and let F_{1} = F − x_{1}δ_{1}. It follows that |^{∗}F_{1}|(p) ≈ ∥^{∗}F_{1}(p_{1})∥ = ∥^{∗}F (p_{1}) − st(F (p_{1}))∥ ≈ 0. Further, follows from the transfer principle, that |F_{1}| ⩽ |F |.

Iterating this process we obtain standard sequences (F_{n})_{n∈ℕ}, (x_{n})_{n∈ℕ}, and (δ_{n})_{n∈ℕ} such that F_{n} = F − ∑_{i=1}^{n} x_{i}δ_{i}, |F_{n+1}| ⩽ |F_{n}| ⩽ |F |, and ∥x_{n+1}∥ ⩽ ∥x_{n}∥ for each n ∈ ℕ. Let F̄ = F − ∑_{i=1}^{∞} x_{i}δ_{1}, then an argument similar to the one in the proof of Theorem 5.1 shows that F̄ is strongly continuous. □

6. Horn-Tarsky Theorem

6.1. Theorem. Let A be an algebra of subsets of a set X, C a subalgebra of A, and μ a finitely additive measure on C. Then μ can be extended to a finitely additive measure on A.

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Proof. Let P_{A} and P_{C} be hyperfinite ifp’s for A and C respectively. Without loss of generality we can assume P_{A} is a refinement of P_{C}. Take p ∈ P_{C} and let p_{1}, p_{2}, …, p_{N} be a hyperfinite partition of p into elements of P_{A}. Next, we assign a weight w(p_{i}) ∈ ^{∗}ℝ_{+} to each p_{1} as 1 ⩽ i ⩽ N so that ∑_{i=1}^{N} w(p_{i}) = ^{∗}μ(p).

We apply this procedure to every p ∈ P_{C}. Now each element of P_{A} is assigned a weight. For a set A ∈ A we put

λ(A) = st ( ∑_{p∈P_{A}, p⊆^{∗}A} w(p)).

It can be easily verified that λ is a standard finitely additive measure and λ_{|C} = μ. □

Obviously, the same reasoning can be used to prove Horn-Tarsky Theorem for a Banach-valued measure, but again we would need the range of the measure be relatively compact so that we could take standard parts when defining λ.

7. Ergodic transformations

Let (X , A) again be a measurable space, and let τ: X → X be a measurable transformation, i.e., τ^{−1}(A) ∈ A for each A ∈ A. We say that a set A is τ-invariant if τ^{−1}(A) = A. If P is an ifp for (X , A), we say that P is τ-invariant if τ^{−1}(p) ∈ P for each p ∈ P. In this case τ^{−1} induces a permutation of elements of P. Let μ be a probability measure on (X , A). Recall that a measurable transformation τ is said to be measure-preserving if μ(τ^{−1}(A)) = μ(A) for each A ∈ A, and ergodic if each τ-invariant set has measure zero or one. If P is an ifp, we write P_{+} = {p ∈ P | ^{∗}μ(p) > 0}.

7.1. Proposition. If τ is ergodic and measure preserving, and P is a τ-invariant ifp, then P_{+} is hyperfinite and τ-invariant, all the elements of P_{+} have the same measure, and τ^{−1} induces a cyclic permutation of elements of P_{+}.

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Proof. Fix p ∈ P_{+} and consider the sequence {τ^{−k}(p) | k ∈ ^{∗}ℕ }, i.e., the orbit of p in P under τ^{−1}. Since τ is measure-preserving, all the sets in the orbit have the same measure. If all of them were distinct, hence disjoint, this would contradict to μ being a probability measure. Thus, we have τ^{−n}(p) = τ^{−m}(p) for some distinct m and n. Without loss of generality we can assume that m = 0, i.e., τ^{−n}(p) = p for some n ∈ ^{∗}ℕ. Then A = ⋃_{k=0}^{n−1} τ^{−k}(p) is a τ-invariant set of positive measure. Since τ is ergodic, we have μ(A) = 1. This implies that every element of P_{+} is contained in A, so that P_{+} = {τ^{−k}(p) | k = 0, …, n − 1}. It follows immediately that P_{+} is hyperfinite and that τ acts as a cyclic permutation of the elements of P_{+}. □

Unfortunately, one cannot always find an invariant ifp. For example, it is easy to see that no ifp is invarial under a strongly mixing transformation. Recall, that a measure-preserving transformation τ is called a strong mixing if μ(A ∩ τ^{−n}(B)) → μ(A)μ(B) for any two A, B ∈ A. Indeed, if P were a τ-invariant ifp, then by Proposition 7.1 P has to be hyperfinite, and τ^{−1} induces a cyclic permutation of the elements of P_{+}. But then μ(p_{1} ∩ τ^{−n}(p_{2})) does not converge for any p_{1}, p_{2} ∈ P, a contradiction.

References

[BRBR83] K. P. S. Bhaskara Rao and M. Bhaskara Rao. Theory of charges. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1983. A study of finitely additive measures, With a foreword by D. M. Stone, MR86f:28006.

[KM89] A. G. Kusraev and S. A. Malyugin. Atomic decomposition of vector measures. Sibirsk. Mat. Zh., 30(5):101–110, 216, 1989. MR90k:28022.

[Loe72] Peter A. Loeb. A non-standard representation of measurable spaces, L_{∞}, and L_{∞}^{∗}. pages 65–80. Studies in Logic and Found. Math., Vol. 69, 1972. MR58:2215.

[Sun92] Yeneng Sun. On the theory of vector valued Loeb measures and integration. J. Funct. Anal., 104(2):327–362, 1992. MR93a:46146.

[Tro93] Vladimir G. Troitsky. On some problems of nonstandard measure theory. Master’s thesis, Novosibirsk State University, 1993.

  E-mail address: vladimir@math.utexas.edu