1F.3] 1F. Countable operations ... 35

and for each ordinal number 1, let

Σ0

= ¬(

Σ0).

Unscrambling this, P is in Σ0 is there are pointsets P0,P1,... with each Pi in some

Σ0,

, such that

P =

i

(X \ Pi).

We call

Σ0

the Borel pointclass of order . The dual and ambiguous Borel pointclasses

are defined in the obvious way,(8,9)

Π0

=

¬Σ0,

∆0

=

Σ0

∩

Π0.

For finite this definition yields the pointclasses Σ0 as we know them, so there is no

conflict in notation.

It is very easy to extend the basic properties of the finite Borel pointclasses to all

Borel pointclasses and we will leave this for the exercises. We only state here the basic

characterization of the pointclass B of Borel

sets,(8)

B = Σ0.

1F.3. Theorem. For each product space X the class B X of Borel subsets of X is

the smallest collection of subsets of X which contains the open sets and is closed under

complementation and countable union; similarly, B X is the smallest collection of

subsets of X which contains the open (or the closed) sets and is closed under countable

union and countable intersection.

Proof. If P is Borel, then P is in Σ0 for some , so ¬P = X \ P ∈ Σ0+1, in

particular, ¬P is Borel. Also, if Pi is Borel for every i, Pi ⊆ X, then Pi ⊆

Σ0

i

for

some i,so ¬Pi ∈

Σ0

i

+1

and taking

= supremum{

i

+2 : i = 0,1,2,...},

we have P ∈

Σ0,since

P =

i

Pi =

i

(

X \ (X \ Pi)

)

.

Thus the class of Borel subsets of X is closed under ¬ and .

Conversely, if S is any collection of subsets of X which is closed under ¬ and ,

then S clearly contains all open subsets of X and an easy induction on shows that

P ⊆ X,P ∈

Σ0

=⇒ P ∈ S.

For the second assertion, notice first that B X is easily closed under countable

intersection, since

i

Pi = X \

(

i

(X \ Pi)

)

.

Conversely, if S contains all the opensubsets of X and is closedunderboth countable

union and countable intersection, then each P ⊆ X which is either

Σ0

or

Π0

is in S

byatrivialinductionon ; becauseclosedsetsarecountableintersectionsofopensets

and in general

P ∈

Σ0

=⇒ P =

i

Pi with each Pi ∈

Π0

i

,

i

,

P ∈

Π0

=⇒ P =

i

Pi with each Pi ∈ Σ0

i

,

i

.