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METRIC SPACE VALUED RANDOM VARIABLES
CLINTON T. CONLEY
Given a bounded metric space (V, d) and a probability space (X, μ), we denote by MALG(X, μ) the σ-algebra of μ-measurable subsets of X (taken modulo null sets), and we denote by L(X, μ, V ) the space of V -valued random variables, i.e., the set of μ-measurable functions from X to V. After making the standard identification between functions agreeing μ-almost everywhere, L(X, μ, V ) may be equipped with the metric d˜(f, g) =
d(f (x), g(x)) dμ(x). For v ∈ V , let cv ∈ L(X, μ, V ) denote the constant v function (so cv (x) = v for all x ∈ X), and let CV = {cv : v ∈ V }. Henceforth, we fix a standard probability space (X, μ). Let add(null) denote the smallest cardinal κ for which there exists a sequence (Aα)α<κ of μ-null subsets of X such that
α<κ Aα^ is not^ μ-null. If^ A ⊆^ MALG(X, μ) is a sub-σ-algebra, we denote by μA the restriction of μ to A. Observe that the measure algebra of (X × X, μ × μA) is separable and non-atomic, and is therefore isomorphic to MALG(X, μ) (see, e.g., Exercise 17.44 in [1]). If π is an isomorphism witnessing this, and if (W, d) is a bounded metric space with |W | < add(null), we have an isometry f 7 → f˜ between L(X × X, μ × μA, W ) and L(X, μ, W ) defined by
f˜ −^1 (w) = π(f −^1 (w)).
We immediately see the following.
Theorem 1. Suppose that (V, dV ) and (W, dW ) are bounded metric spaces with |W | < add(null), and suppose there exists some sub-σ-algebra A ⊆ MALG(X, μ) such that V ∼= L(X, μA, W ). Then L(X, μ, V ) ∼= L(X, μ, W ).
Proof. Fix a sub-σ-algebra A ⊆ MALG(X, μ) such that V is isometric to L(X, μA, W ). We see by Fubini that
L(X, μ, V ) ∼= L(X, μ, L(X, μA, W )) ∼= L(X × X, μ × μA, W ) ∼= L(X, μ, W ),
completing the proof. 2
The main focus of this paper is to examine when the converse of Theorem 1 holds. Given any metric space (W, d), we may associate with each w ∈ W a partial order ≤w of W defined by
w 1 ≤w w 2 ⇔ d(w 1 , w) + d(w 1 , w 2 ) − d(w 2 , w) = 0.
Let us confirm that ≤w is transitive. Suppose that w 1 , w 2 , w 3 ∈ W are such that w 1 ≤w w 2 and w 2 ≤w w 3. Twice using the triangle inequality, we see
d(w 1 , w) + d(w 1 , w 3 ) ≥ d(w 3 , w) = d(w 2 , w) + d(w 2 , w 3 ) = d(w 1 , w) + d(w 1 , w 2 ) + d(w 2 , w 3 ) ≥ d(w 1 , w) + d(w 1 , w 3 ),
so d(w 1 , w) + d(w 1 , w 3 ) = d(w 3 , w) and hence w 1 ≤w w 3. We may view w 1 ≤w w 2 as expressing that w 1 “lies on a geodesic” between w and w 2. For convenience of notation, we fix some element 0 ∈ W and suppress the subscript in ≤ 0. If (W, d) is a bounded metric space, we use the symbol (^4) w to denote the partial order ≤cw of L(X, μ, W ). Again, when there is no danger of confusion we write 4 instead of 40. It is immediate that w 1 ≤w w 2 if and only if cw 1 4 w cw 2. The following proposition generalizes this fact.
Proposition 2. Suppose that (W, d) is a bounded metric space, w ∈ W , and f, g ∈ L(X, μ, W ). Then f (^4) w g if and only if f (x) ≤w g(x) on a set of full measure.
Proof. Note that q : x 7 → d(f (x), w) + d(f (x), g(x)) − d(g(x), w) is a non-negative function. Thus,
q(x) dμ(x) = 0 if and only if ∀μx (q(x) = 0). Then
f (^4) w g ⇔ d˜(f, cw) + d˜(f, g) − d˜(g, cw) = 0
⇔
q(x) dμ(x) = 0
⇔ ∀μx (q(x) = 0) ⇔ ∀μx (f (x) ≤w g(x)). 2
Given a poset (P, ≤P ), we say that a set A ⊆ P is closed under meets in (P, ≤P ) if for all sequences (ai)i∈I in A with |I| < add(null), if
i∈I ai^ exists in^ P^ then it is in A. Analogously, we say that A is closed under joins in (P, ≤P ) if for all sequences (ai)i∈I in A with |I| < add(null), if
i∈I ai^ exists in^ P^ then it is in^ A.
Proposition 3. Suppose that (W, d) is a bounded metric space. Then for all w ∈ W , the set of constant functions is closed under meets and joins in (L(X, μ, W ), (^4) w).
Proof. Without loss of generality, we may assume w = 0 and work with the partial order 4. We show that the set CW = {cw : w ∈ W } is closed under meets in (L(X, μ, W ), 4 ). The argument that C is closed under joins is similar. Fix a sequence (wi)i∈I , |I| < add(null), of elements of W , and suppose that f =
i∈I cwi exists in^ L(X, μ, W^ ). By Proposition 2, we know that for all^ i^ ∈^ I, ∀μx (f (x) ≤ wi). Since |I| < add(null), this implies that ∀μx ∀i (f (x) ≤ wi). Equivalently, ∀μx ∀i (cf (x) 4 cwi ). Since f is the greatest lower bound of (cwi )i∈I , we have that ∀μx (cf (x) 4 f ). Upon appealing once more to Proposition 2, we see that ∀μx ∀μy (f (x) ≤ f (y)), implying that f is constant on a set of full measure. 2
Given a metric space (W, d) and w 0 , w 1 ∈ W , we let w 0 w 1 denote the set {w ∈ W : w ≤w 0 w 1 }, that is, the set of points between w 0 and w 1. We say that w 2 ∈ W discerns the endpoints of w 0 w 1 if w 2 6 ∈ w 0 w 1 and, for i ∈ { 0 , 1 },
∀w ∈ w 0 w 1 \ {wi} (w ∧w 2 wi = w 2 ).
We say that a bounded metric space W is L-minimal if whenever L(X, μ, W ) ∼= L(X, μ, V ), there exists an isometry ϕ : L(X, μ, W ) → L(X, μ, V ) such that ϕ[CW ] ⊆ CV. For example, all sets of cardinality less than add(null), equipped with the dis- crete metric, are L-minimal.
Theorem 5. Suppose that (V, dV ) is a bounded metric space, and suppose that (W, dW ) is a bounded, self-discerning, L-minimal metric space with |W | < add(null). Then the following are equivalent:
- V is isometric to L(X, μA, W ) for some sub-σ-algebra A ⊆ MALG(X, μ);
- L(X, μ, V ) is isometric to L(X, μ, W ).
Proof. (1) ⇒ (2) is simply an instance of Theorem 1. (2) ⇒ (1): Fix an isometry ϕ : L(X, μ, W ) → L(X, μ, V ) such that ϕ[CW ] ⊆ CV. Let F = ϕ−^1 (CV ), so F ∼= V. We have that F contains CW and, by Proposition 3, F is closed under meets and joins. Thus, by Proposition 4, there exists a sub-σ-algebra A ⊆ MALG(X, μ) such that F ∼= L(X, μA, W ) as desired. 2
Corollary 6. Suppose that V and W satisfy the hypotheses of Theorem 5. Suppose in addition that W is a metric group. If L(X, μ, V ) ∼= L(X, μ, W ), then V can be given a group structure compatible with its metric.
Proof. By Theorem 5, we have that V ∼= L(X, μA, W ) for some sub-σ-algebra A ⊆ MALG(X, μ). V then forms a metric group under the operations of pointwise multiplication and inversion. 2
If we equip { 0 , 1 } with the discrete metric, it is easily seen that L(X, μ,{0,1}) ∼= (MALG(X, μ), dμ), where dμ(A, B) = μ(A 4 B). As mentioned, { 0 , 1 } is L-minimal, but it fails to be self discerning. Nevertheless, the analogue of Theorem 5 remains true.
Theorem 7. Suppose that (V, d) is a bounded metric space. The following are equivalent:
- V is isometric to the restriction of MALG(X, μ) to a sub-σ-algebra;
- L(X, μ, V ) is isometric to MALG(X, μ).
Proof. (1) ⇒ (2) is yet another instance of Theorem 1. (2) ⇒ (1): Fix an isometry ϕ : L(X, μ, V ) → MALG(X, μ). We may assume ϕ(c 0 ) = ∅: if this is not the case, we replace ϕ by ϕ′^ defined by ϕ′(f ) = ϕ(f ) 4 ϕ(c 0 ). We will show that A = {ϕ(cv ) : v ∈ V } forms a sub-σ-algebra of MALG(X, μ). Since v 7 → ϕ(cv ) is an isometry between V and A, this will complete the proof of the theorem. We first notice that if we have such a ϕ, then for each v ∈ V there is a unique v ∈ V with d(v, v) = 1. To see existence, we consider f = ϕ−^1 (X \ ϕ(cv )). Since
d^ ˜(cv , f ) = dμ(ϕ(cv ), ϕ(f )) = 1, we may find a v ∈ V with d(v, v) ≥ 1. Of course, we must have that the diameter of V is at most 1, so d(v, v) = 1. On the other hand, if v 1 and v 2 are such that d(v, v 1 ) = d(v, v 2 ) = 1, then dμ(ϕ(cv ), ϕ(cv 1 )) = dμ(ϕ(cv ), ϕ(cv 1 )) = 1. Thus, ϕ(cv 1 ) and ϕ(cv 2 ) are both com- plements of ϕ(cv ), and hence equal, so v 1 = v 2. Since dμ(ϕ(cv ), ϕ(cv )) = 1, we conclude that A is closed under complements.
To see A is closed under countable intersections, we simply observe that ϕ induces an isomorphism between the posets (L(X, μ, V ), 4 ) and (MALG(X, μ), ⊆). Then, since MALG(X, μ) is closed under countable intersections, Proposition 3 implies that A is as well. 2
Corollary 8. Suppose that (W, d) is a bounded metric space such that |W | < c. If L(X, μ, W ) is isometric to MALG(X, μ), then W is finite.
Proof. The smallest infinite sub-σ-algebra of MALG(X, μ) has size c. 2
Regrettably, there is not yet a simple characterization of metric spaces which are both L-minimal and self discerning. Additionally, in light of Theorem 7, it appears that the definition of a self discerning space is a bit too restrictive, as it excludes the smallest nontrivial space for which the theorem is true.
References
[1] A.S. Kechris. Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
