Zeitschr. f . math. Lo& und Orundlagen d . Math. Bd. 33, S , 199-200 (1987)

SOME PROPERTIES OF THIN I7:,+, SETS

by YUTAKA YASUDA in Tokyo (Japan)')

For unexplained notation and terminology see MOSCHOVAKIS [3]. In particular, we use T X , p, y for reals (as usual, total functions from w to o) and 6, A for ordinals. We work in set theory ZF + DC + Det(d:,), and we shall need the following basic theorems.

= qrl([), [ < A, is null. Then if the associated prewellordering with p is Lebesgue measurable, P is also null.

( I ) Let p: P + A be a norm on P each of whose levels

(11) (MARTIN, MOSCHOVAKIS) Every IT^,,,, set admits a I7&,+,-norm.

(111) (KECHRIS) There is the largest thin (i.e. containing no nonempty perfect sub-

(IV) (KECHRIS) 174n+ sets are Lebesgue measurable.

(V) (MARTIN) If P is a countable L';,,+l(a) set then every element of P is a A:,,+,(a)

All these results can be found in MOSCHOVAKIS [3].

Theorem 1. Let P be a givenI7~,+, set of reals and p: P -++ A any II&,+,-norrn on P. Put Pp = { a E P,": (V/? E Pr) ( a ~ d ~ , + , ( ~ ) ) } . Then u{@: 6 < A } is a thin I7:,+, subset of P.

set) n:,,, set C2,+,.

real.

Proof . Let

w, 9) = {. E p : vpw, p E p p(/?) = da) * a E &,+,(p))} >

a s p - a, p E w, 9) & 944 s d p ) . . Then U{pp: E < A} = S(P, 9) is a 171,+1 subset of P , and 5 is a lTi,+l prewell- ordering on (J{pr: 6 < A} each of whose levels (this is one of 151's) is countable. Now we must show that U{&: 6 < A} is thin. Toward a contradiction, suppose that there is a nonempty perfect subset K of u{f'F: [ < A}, let f be a continuous one-to-one mapping from "2 to K . Let

a s * p o a , / 3 ~ " 2 & t ( a ) sj(/3). Then s* is aI7:,+, prewellordering on '"2 each of whose levels is null. Therefore, by (I) and (IV), "2 must be null This is a contradiction. 0

Corol lary 2 (GUASPARI [l]) . If 5 i s a I7:,+, wellordpring on a set of reals, then its field i s a thin 17:,+, set.

Proof . Let P = { a : CI 5 a), the field of 5 , and p: ( P , 5 ) H) (A, s ) the order- isomorphism. Then P is a set and p is a I7.&,+,-norm on P each of whose

I ) I thank Professor HISAO TANAKA for his constant encouragement and valuable aid in the preparation of this paper.

200 YU. YASUDA

levels is singleton. Therefore, PI = P f for 6 < A. By theorem 1, the field of 5 P = U{pF: E < A} is a thin h';,,+l set. 0

Theorem 3 (KECHRIS [2] for direction). Let cp: P -H A be a Il~,+,-nor?rr 012. /L

set P. Then P is thin if and only if (Va, @ E P ) ( y ( a ) 5 p(p) = a E A ; , , + ~ ( / ~ ) ) . Proof . For * direction: Toward a contradiction, suppose that there are reals x o

and Po in P such that cp(ao) 5 cp(po) and a. #O:,,+,(~,). Then { y E P : p ( y ) =< y( / jo) l is a Zi,+,(,80) subset of P and the real a, ( # O ~ , + , ( ~ o ) ) is in this set. By (V), P rnust have a nonempty perfect subset. We have a contradiction. For -+ direction: Since a? = PF for 6 < A, by theorem 1 P = U { P f : 5 < A} is a thin set. 0

Theorem 4. Let Q be a 17~,,+, set and R a Xi,,+, set. Then i f Qnh' is countrcOl~ it i s a subset of C2n+l .

Proof . Let AS, = { a E Q A R: cp(a) 5 cp(p)), where cp is a n:,,,-norm on Q Thcn if p E Q, ~7~ is a countable Zi,,+,(@) set, so by (V) every element of ASB is a /l;,ttl([j) rcal. Now let a E Q A R and cp(a) = ~ ( p ) . Then a E S,, so M E A : , + , ( P ) . This means Q A R E U{Q,": 6 < A} E C,,,, , where A is the length of the norm cp.

From theorem 4 we have

Corol la ry 5 . Let Q, be an: ,+ , (a ) sd and R, a Zin+l(a) set for each i < {I). Then if U{(Qi n R , ) : i < a} is countable it i s a subset of C2,,t1(a). 0

Theorem 6 (STEEL and MARTIN for n = 0, see MOSCHOVAKIS [3]). There i s a Zh,,+l sPt which cannot be uniformized by a (Zin+,)Qa set.

Proof . Let P(a, @) o p # C,,,+,(a). Then P is in Xf,,+, and Va 3@P(a, p) Towards a contradiction, suppose that P can be uniformized by a (Z;,,+l)ou set P*. Then P*(a, p ) has the form 3 i ( & ( i , a, p) &R(i , a, p ) ) , where Q is in 17i,+,(a*) and R is in Z;,+,(a*) for some fixed a*. Let p* be the unique real such that, 3 i ( Q ( i , x * , p*) 1G & R(i, a*, p*)) holds. By Corollary 5

(@*} = u { p : Q(i, a*, PI tk R(i, a*, P I } s C2,,+1(a*), i < W

i.e. p* is in C,,,+,(a*). This is a contradiction. 0

References

[l] GUASPARI, D., Thin and wellordered analytical sets. Ph. D. thGeis, Cambridge University 1973. [2] KECHRIS, A. S., Projective ordinals and countable analytical sets. Ph. D. thesis, UCLA 1972. [3] P r l o s c ~ o v ~ ~ r s , Y. N., Descriptive Set Theory. North-Holland Publ. Comp., Amsterdam 19x0.

Yntaka Yasnda 22-14-1209 Minami-Aoxama 2-Chome Minato-Ku, Tokyo 107 Japan

(Eingegangen am 14. Februar 1986)