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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.

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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

σ

set. In particular, M ∩ F

σ

is a base for M.

Proof. Let A be a meagre subset of a topological space. Then A = S

n=1

A

n

for some nowhere dense sets A

n

. The set S

n=1

A

n

, which contains A, is meagre and F

σ

since the sets A

n

are nowhere dense and closed.

Proposition 1.3 Each null subset of R is contained in some null G

δ

set. In other words, N ∩ G

δ

is a base for N .

Proof. This is immediate from the regularity of the Lebesgue measure.

Proposition 1.4 Every uncountable G

δ

subset of R contains a nowhere dense null closed set with cardinality 2

ω

.

Proof. Let G be an uncountable G

δ

set. Then G = T

n=0

U

n

for some open sets U

n

. We may construct a Cantor scheme { I

s

| s 2

} such that I

s

is a compact nonde- generate interval contained in U

|s|

with µ(I

s

) 5 3

−n

for every s 2

. 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

n

3

n

0 as n → ∞.

new@ms.u-tokyo.ac.jp

1

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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

, 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

n

for some nowhere dense sets A

n

. Since S

n=1

A

n

is a meagre F

σ

set that contains A

c

, the set ( S

n=1

A

n

)

c

is a nonmeagre G

δ

set contained in A. Therefore Proposition 1.4 shows that ( S

n=1

A

n

)

c

contains 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

σ

subset A and a null G

δ

subset 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

n

and A = B

c

. Since P

n

is open for every positive integer n, we have B ∈ G

δ

and 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

nc

is nowhere dense because P

n

is open and dense. It follows from A = S

n=1

P

nc

that 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

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Proof. Since |M ∩ F

σ

| = 2

ω

and |N ∩ G

δ

| = 2

ω

, it follows from Proposition 2.1 that there exist bijections ξ 7−→ A

ξ

from 2

ω

to M ∩ F

σ

, and ξ 7−→ B

ξ

from 2

ω

to N ∩ G

δ

that satisfy A

0

B

0

= and A

0

B

0

= R.

Define inductively a map ξ 7−→ F

ξ

from 2

ω

to M such that (1) F

0

= A

0

;

(2) F

ξ+1

is the union of F

ξ

∪A

ξ

and a meagre set contained in (F

ξ

∪A

ξ

)

c

with cardinality 2

ω

for every ξ 2

ω

;

(3) F

ξ

= S

α∈ξ

F

α

for 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

ξ

from 2

ω

to N such that (1) G

0

= B

0

;

(2) G

ξ+1

is the union of G

ξ

B

ξ

and a null set contained in (G

ξ

B

ξ

)

c

with cardinality 2

ω

for every ξ 2

ω

;

(3) G

ξ

= S

α∈ξ

G

α

for 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

ξ

since |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

0

to S

ξ∈2ω

(G

ξ+1

\ G

ξ

) = R \ G

0

= F

0

. Thus the union f of f e and f e

−1

is an involution from R to R.

Let M be a meagre set. Proposition 1.2 implies that M is contained in A

ξ0

for some ξ

0

2

ω

. Then

M A

ξ0

F

ξ0+1

= [

ξ∈ξ0+1

(F

ξ+1

\ F

ξ

) F

0

shows 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

(G

ξ+1

\ G

ξ

) G

0

= G

ξ0+1

∈ N , which implies that f(M ) is null.

Similarly it follows from Proposition 1.3 that f (N ) is meagre for every null set N.

Since f is an involution, we conclude that f (M ) is null only if M is meagre, and that f(N ) is meagre only if N is null.

Remark. Assuming the continuum hypothesis is too much; the proof of Theorem 2.2 works on the mere assumption that add(M) = add(N ) = 2

ω

. In particular, assuming the Martin axiom is enough.

3

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3 References

[1] Tomek Bartoszy´nski and Haim Judah, Set Theory—On the Structure of the Real Line, A K Peters.

[2] Winfried Just and Martin Weese, Discovering Modern Set Theory II, Graduate Studies in Mathematics, 18, American Mathematical Society.

[3] John C. Oxtoby, Measure and Category, Graduate Texts in Mathematics, 2, Springer- Verlag.

[4] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, 180, Spring-Verlag.

4

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