The Erd¨os-Sierpi´nski Duality Theorem
Shingo SAITO ∗
0 Notation
The Lebesgue measure on R is denoted by µ. The σ-ideals that consist of all meagre subsets and null subsets of R are denoted by M and N respectively.
1 Similarities between Meagre Sets and Null Sets
Definition 1.1 Let I be an ideal on a set. A subset B of I is called a base for I if each set in I is contained in some set in B.
Proposition 1.2 Each meagre subset of a topological space is contained in some meagre F
σset. In particular, M ∩ F
σis a base for M.
Proof. Let A be a meagre subset of a topological space. Then A = S
∞n=1
A
nfor some nowhere dense sets A
n. The set S
∞n=1
A
n, which contains A, is meagre and F
σsince the sets A
nare nowhere dense and closed.
Proposition 1.3 Each null subset of R is contained in some null G
δset. In other words, N ∩ G
δis a base for N .
Proof. This is immediate from the regularity of the Lebesgue measure.
Proposition 1.4 Every uncountable G
δsubset of R contains a nowhere dense null closed set with cardinality 2
ω.
Proof. Let G be an uncountable G
δset. Then G = T
∞n=0
U
nfor some open sets U
n. We may construct a Cantor scheme { I
s| s ∈ 2
<ω} such that I
sis a compact nonde- generate interval contained in U
|s|with µ(I
s) 5 3
−nfor every s ∈ 2
<ω. Let f : 2
ω−→ R denote the associated map of the Cantor scheme defined by ©
f (α) ª
= T
∞n=1
I
α|n. Denote the range of f by A.
Note that A = T
∞n=1
S
s∈2n
I
s, which implies that A is closed. Moreover A is null because
µ(A) 5 µ Ã [
s∈2n
I
s!
5 X
s∈2n
µ(I
s) 5 2
n3
n→ 0 as n → ∞.
∗
new@ms.u-tokyo.ac.jp
1
Thus Int A = ∅, which shows that A is nowhere dense since A is closed. The injectivity of f shows that |A| = 2
ω. Since I
s⊂ U
|s|for every s ∈ 2
<ω, we have
A =
\
∞n=1
[
s∈2n
I
s⊂
\
∞n=1
U
n= G.
Corollary 1.5 Every residual subset of R contains a meagre set with cardinality 2
ω. Proof. Let A be a residual subset of R. Then A
c= S
∞n=1
A
nfor some nowhere dense sets A
n. Since S
∞n=1
A
nis a meagre F
σset that contains A
c, the set ( S
∞n=1
A
n)
cis a nonmeagre G
δset contained in A. Therefore Proposition 1.4 shows that ( S
∞n=1
A
n)
ccontains a meagre set with cardinality 2
ω, which is contained in A.
Corollary 1.6 The complement of each null subset of R contains a null set with cardi- nality 2
ω.
Proof. Let A be a subset of R with A
c∈ N . Then A is measurable and µ(A) = ∞.
Therefore the regularity of µ yields a closed set F contained in A with µ(F ) = ∞. It follows from Proposition 1.4 that F contains a null set with cardinality 2
ω, which is contained in A.
2 Erd¨ os-Sierpi´ nski Duality Theorem
Proposition 2.1 There exist a meagre F
σsubset A and a null G
δsubset B of R that satisfy A ∩ B = ∅ and A ∪ B = R.
Proof. Enumerate Q = {q
1, q
2, . . .}, and put P
n= S
∞j=1
B(q
j, 2
−n−j) for positive inte- gers n. Define B = T
∞n=1
P
nand A = B
c. Since P
nis open for every positive integer n, we have B ∈ G
δand A ∈ F
σ. We shall prove that A is meagre and B is null.
The set B is null because µ(B) 5 µ(P
n) 5
X
∞j=1
µ ¡
B(q
j, 2
−n−j) ¢
= X
∞j=1
2
−n−j+1= 2
−n+1→ 0 as n → ∞.
For every positive integer n, the set P
ncis nowhere dense because P
nis open and dense. It follows from A = S
∞n=1
P
ncthat A is meagre.
Theorem 2.2 (Erd¨ os-Sierpi´ nski Duality Theorem) Assume that the continuum hy- pothesis holds. Then there exists an involution f : R −→ R such that f (A) is meagre if and only if A is null, and f (A) is null if and only if A is meagre for every subset A of R.
2
Proof. Since |M ∩ F
σ| = 2
ωand |N ∩ G
δ| = 2
ω, it follows from Proposition 2.1 that there exist bijections ξ 7−→ A
ξfrom 2
ωto M ∩ F
σ, and ξ 7−→ B
ξfrom 2
ωto N ∩ G
δthat satisfy A
0∩ B
0= ∅ and A
0∪ B
0= R.
Define inductively a map ξ 7−→ F
ξfrom 2
ωto M such that (1) F
0= A
0;
(2) F
ξ+1is the union of F
ξ∪A
ξand a meagre set contained in (F
ξ∪A
ξ)
cwith cardinality 2
ωfor every ξ ∈ 2
ω;
(3) F
ξ= S
α∈ξ
F
αfor every limit ordinal ξ ∈ 2
ω.
We may construct such a map due to Corollary 1.5 and the continuum hypothesis.
Similarly Corollary 1.6 and the continuum hypothesis allow us to define a map ξ 7−→
G
ξfrom 2
ωto N such that (1) G
0= B
0;
(2) G
ξ+1is the union of G
ξ∪ B
ξand a null set contained in (G
ξ∪ B
ξ)
cwith cardinality 2
ωfor every ξ ∈ 2
ω;
(3) G
ξ= S
α∈ξ
G
αfor every limit ordinal ξ ∈ 2
ω.
For each ξ ∈ 2
ω, there exists a bijection f
ξ: F
ξ+1\ F
ξ−→ G
ξ+1\ G
ξsince |F
ξ+1\F
ξ| =
|G
ξ+1\ G
ξ| = 2
ω. Put f e = S
ξ∈2ω
f
ξ. Then f e is a bijection from S
ξ∈2ω
(F
ξ+1\ F
ξ) = R \ F
0= G
0to S
ξ∈2ω
(G
ξ+1\ G
ξ) = R \ G
0= F
0. Thus the union f of f e and f e
−1is an involution from R to R.
Let M be a meagre set. Proposition 1.2 implies that M is contained in A
ξ0for some ξ
0∈ 2
ω. Then
M ⊂ A
ξ0⊂ F
ξ0+1= [
ξ∈ξ0+1
(F
ξ+1\ F
ξ) ∪ F
0shows that
f (M) ⊂ f à [
ξ∈ξ0+1
(F
ξ+1\ F
ξ) ∪ F
0!
= [
ξ∈ξ0+1
f (F
ξ+1\ F
ξ) ∪ f (F
0)
= [
ξ∈ξ0+1