Seria I: PRACE MATEMATYCZNE XLV (1) (2005), 125-129
I.V. Shragin
Some remarks on Carath´ eodory conditions
Abstract. Let (T , T) be a measurable space, X and Y be metric spaces. For the maps from T × X into Y the notions of supmeasurability, standardness and Carath´eodory conditions are considered, and the logical relations between these notions are dis- cussed.
2000 Mathematics Subject Classification: 28A20.
Key words and phrases: supmeasurable function, standard function, Carath´eodory function.
Let (T , T) be a measurable space, X and Y be metric spaces, B(X ) and B(Y) be the σ-algebras of their Borel sets. Denote by Λ the product T × B(X ), i.e. the σ-algebra on T × X generated by the family {A × B : A ∈ T, B ∈ B(X )}. With each function ϕ : T → X we associate its ”graph-function” Gϕ: T → T × X , where Gϕ(t) = (t, ϕ(t)) (so that Gϕ(T ) = grϕ).
Lemma 1 A function ϕ : T → X is (T, B(X ))-measurable (i.e. ∀B ∈ B(X ) ϕ−1(B) ∈ T) if and only if Gϕis (T, Λ)-measurable.
Proof If ϕ is (T, B(X ))-measurable, by A ∈ T, B ∈ B(X ) G−1ϕ (A × B) = {t : (t, ϕ(t)) ∈ A × B} = A ∩ ϕ−1(B) ∈ T, i.e. Gϕ is (T, Λ)-measurable. The converse statement follows from the equality ϕ−1(B) = G−1ϕ (T × B).
Definition 1 A function f : T × X → Y is called supmeasurable if for any (T, B(X ))-measurable function ϕ the composition f ◦ Gϕ : T → Y is (T, B(Y))- measurable.
Remind the notion of standard function ([1], [2]). It is defined with the help of σ-ideal
(1) J := {A ⊂ T : (E ⊂ A) ⇒ (E ∈ T)}
(evidently, J ⊂ T). Denote by R the σ-filter dual to J , i.e.
R:= {T0⊂ T : T \ T0∈ J }.
It is clear that R ⊂ T and T ∈ R.
Definition 2 A function f : T × X → Y is called standard if there is T0∈ R such that the restriction of f onto T0× X is (Λ, B(Y))-measurable.
For convenience of reading we prove the following
Proposition 1 ([1],[2]) If a function f is standard, then it is supmeasurable.
Proof Let ϕ : T → X be a (T, B(X ))-measurable function and take arbitrary C ∈ B(Y). Then (f ◦ Gϕ)−1(C) = G−1ϕ [f−1(C)]. Let T0be a set from Definition 2 and put
P = f−1(C) ∩ (T0× X ), Q = f−1(C) \ (T0× X ).
As (by Definition 2) P ∈ Λ, by virtue of Lemma 1 G−1ϕ (P ) ∈ T. Further, as Q ⊂ (T \T0)×X , G−1ϕ (Q) ⊂ T \T0∈ J , whence G−1ϕ (Q) ∈ T. Thus (f ◦Gϕ)−1(C) = G−1ϕ (P ∪ Q) ∈ T, i.e. the function f is supmeasurable.
Definition 3 A function f : T × X → Y is called Carath´eodory function (shortly:
C-function) if the following conditions (Carath´eodory conditions) are fulfilled:
1) (∀ x ∈ X ) the function f (·, x) : T → Y is (T, B(Y))-measurable;
2) (∃ T0∈ R) (∀ t ∈ T0) the function f (t, ·) : X → Y is continuous.
In this paper we investigate the relations between the notions of standard function and Carath´eodory function.
Proposition 2 ([1],[2]) Let the space X (with metric d) be separable. Then every C-function f is standard (and ,by virtue of Proposition 1, is supmeasurable).
Proof Let a countable set V be dense in X . Take an arbitrary closed set F ⊂ Y and put
Un= {y ∈ Y : ρ(y, F ) < 1/n}, n = 1, 2, . . . (ρ is a metric in Y). If f0= f |(T
0×X), where T0 is the set from Definition 3, then ([2], Theorem 3)
f0−1(F ) =
∞
\
n=1
[
v∈V
{t ∈ T0: f (t, v) ∈ Un} × {x : d(x, v) < 1/n}.
Hence f0−1(F ) ∈ Λ, i.e. the function f is standard. In connection with Proposition 2 the question arises about essentiality of sepa- rability of X . As we shall see, this condition is not necessary for standardness of some C-functions. We begin at following simple example.
Example 1 Let f (t, x) = gk(x), t ∈ Tk∈ T, k = 1, 2, . . . , x ∈ X , where the sets Tk form a countable (in particular, finite) partition of T , and the functions gk: X → Y are continuous. Then it is easy to check that f is standard C-function by any metric space X .
Note that in Example 1 the standardness of f remains by Borel functions gk. By consideration of another class of standard C-functions we shall use the fol- lowing
Lemma 2 Let T and X be metric spaces, where T is separable. Then B(T × X ) ⊂ B(T ) × B(X ).
Proof Let {Ak} be a countable basis of open sets in T . Then ([3], Proposition 2.2) any open set G in the space T × X has the form G =S
k(Ak× Uk), where Uk are some open sets in X . Hence G ∈ B(T ) × B(X ), from where the statement of the
lemma follows directly.
Proposition 3 Let T , X , Y be metric spaces, where T is separable. Then every C-function f : T × X → Y is standard (here the role of σ-algebra T is played by B(T )).
Proof Let f be C-function. As R ⊂ T = B(T ), the set T0(see Definition 3) is a Borel set in T . Consider the restriction f0of f onto T0×X . By Definition 3 (∀ x ∈ X ) f0(·, x) : T0→ Y is Borel function and (∀ t ∈ T0) the function f0(t, ·) : X → Y is continuous. Then by theorem of Kuratowski ([4], §31, V, Theorem 2) f0is a Borel function,, i.e. (∀C ∈ B(Y)) f0−1(C) ∈ B(T0× X ). As the space T0is separable, by Lemma 2
B(T0× X ) ⊂ B(T0) × B(X ) ⊂ B(T ) × B(X ) = Λ.
Hence f0is (Λ, B(Y))-measurable, i.e. f is standard. So Example 1 and Proposition 3 show that C-function can be standard by non- separable space X . However there exist nonstandard C-functions. For consideration of such example we need the following
Lemma 3 (cf. [5], Exercise 5.1.8) Let (T , T) and (X , B) be measurable spaces.
If a function ψ : T → X has grψ ∈ T × B, then cardψ(T ) ≤ c.
Proof Use the following statement ([5], Exercise 5.1.7): if D ∈ T × B, then the collection {Dt : t ∈ T } of sections Dt := {x ∈ X : (t, x) ∈ D} has at most the cardinality of the continuum. Hence card{(grψ)t : t ∈ T } ≤ c. But since
(grψ)t= ψ(t), then {(grψ)t: t ∈ T } = ψ(T ).
Example 2 Let T = {χE : E ⊂ R} with χE(s) = 1 by s ∈ E, and χE(s) = 0 by s ∈ R \ E. Let σ-algebra T on T consists of the all countable (in particular, finite) subsets of T and their complements. Then countable sets form the σ-ideal J defined in (1).
Indeed, if A ⊂ T and A is countable, then evidently A ∈ J . Let A be a uncount- able set. It is known ([6], Chapter 2, Theorem 13) that there exists E ⊂ A such that card(E)=card(A \ E)=card(A). Hence E 6∈ T and A 6∈ J .
Let X = T . We introduce in X the discrete metric d( d(x1, x2) = 0 by x1 = x2, d(x1, x2) = 1 by x16= x2). Evidently, the space X is nonseparable and all sets in X are open. Hence B(X ) = P(X), where P(X) is the collection of all subsets of X .
Let Y = R with natural metric and f (t, x) = d(t, x). Evidently, the function f : T × X → R is C-function (T plays the role of T0from Definition 3).
For the proof of nonstandardness of f we use Lemma 3. Put ψ(t) = t, t ∈ T . Then card(ψ(T )) =card(T ) = 2c. Hence grψ 6∈ T × B(X ). Note that grψ = ∆, where ∆ is the diagonal in T × T .
Take an arbitrary set T0∈ R (i.e. T \T0is countable) and consider the restriction f0of f onto T0× X . We have
f0−1{0} = {(t, x) ∈ T0× X : x = t} = ∆ ∩ (T0× T0).
Further
(2) ∆ \ (T0× T0) = [
t∈T\T0
({t} × {t}) ∈ T × B(X ) = Λ.
Since ∆ 6∈ Λ, then from (2) it follows that ∆ ∩ (T0× T0) 6∈ Λ. So f0−1{0} 6∈ Λ, i.e. f is nonstandard.
It turns out that the C-function f from Example 2 is supmeasurable, although f is nonstandard. This fact follows from the next account.
First of all we note that if (in general situation described at beginning of the paper) a function f : T × X → Y satisfies the condition 1) from Definition 3 and a function ϕ : T → X is (T, B(X ))-measurable and countably valued (i.e. ϕ(T) is countable), then (as it is easy to check) f ◦ Gϕ is (T, B(Y))-measurable.
Furthermore the following lemma is valid.
Lemma 4 Let T be a uncountable set and σ-algebra T on T consists of all countable sets and their complements. Let further X be a nonempty set. Then if a function ϕ : T → X is (T, P(X ))-measurable, it is countably valued.
Proof Consider the map β : X → T with β(x) = ϕ−1{x}. Obviously, if x16= x2, then β(x1) ∩ β(x2) = ∅.
Assume that (∀ x ∈ X ) β(x) is countable. Since T =S β(x) with x ∈ ϕ(T ) and T is uncountable, then ϕ(T ) is uncountable too. Consider a splitting ϕ(T ) = β1∪β2 with
card(βi) = card(ϕ(T )), i = 1, 2.
Evidently card(ϕ−1(βi)) ≥card(βi), i = 1, 2. So both sets ϕ−1(βi) are uncountable and disjoint with
ϕ−1(β1) ∪ ϕ−1(β2) = T .
Hence ϕ−1(βi) 6∈ T, i = 1, 2, i.e. ϕ is not (T, P(X ))-measurable and that contradicts to hypothesis of the lemma.
Thus by some x0(evidently unique) the set β(x0) is uncountable. As β(x0) ∈ T, T \ β(x0) is countable. Two cases are possible. If β(x0) = T , then ϕ(T ) = {x0}.
And if β(x0) 6= T , then
T \ β(x0) =[
β(x), x ∈ ϕ(T ) \ {x0},
where all β(x) are mutually disjoint. Hence ϕ(T ) is countable in both cases. We return to Example 2 and show that considered C-function f is supmeasurable.
In fact, let ϕ : T → X be a (T, P(X ))-measurable functin. Then, by Lemma 4, ϕ is countably valued. From here, as we noted above, it follows that f ◦ Gϕis (T, B(R))- measurable.
The question remains open about existence of C-function (by nonseparable space X ) that is not supmeasurable.
References
[1] I. V. Shragin, Conditions for measurability of superpositions, Soviet Acad. Sci. Dokl. 197(2) (1971), 295-298 (in Russian) [English translation: Soviet Mathematics, Doklady, 12(2) (1971), 465-470].
[2] I. V. Shragin, Superpositional measurability, Izvestija Vysshikh Uchebnykh Zavedenij, Matem- atika 1 (1975), 82-92 (in Russian).
[3] S. J. Leese, Measurable selections and the uniformization of Souslin sets, Amer. J. Math.
100(1) (1978), 19-41.
[4] K. Kuratowski, Topology, v.1, Academic Press, New York and London 1966.
[5] D. L. Cohn, Measure theory, Birkh¨auser, Boston 1993.
[6] I. Kaplansky, Set theory and metric spaces, Chelsea Publ. Comp., New York 1977.
I.V. Shragin
Florenzen Str. 2-6, 20/2, 50765 K¨oln, Germany E-mail: marina spektor@mail.ru
(Received: 8.12.2004; revised: 28.02.2005)
