projections of analytic sets are analytic

projections of analytic sets are analytic

Projections along compact paved spaces

Given sets

X

and

K

, the

projection map

π

X

:

X

×

K

→

X

is defined by

π

X

⁢

(

x

,

y

)

=

x

. An important property of

analytic sets

(

http://planetmath.org/AnalyticSet2

) is that they are stable under

projections

.

Theorem 1

.

Let

(

X

,

F

)

be a

paved space

(

http://planetmath.org/PavedSpace

),

(

K

,

K

)

be a

compact

(

http://planetmath.org/PavedSpace

) paved space and

π

X

:

X

×

K

→

X

be the projection map.

If

S

⊆

X

×

K

is

F

×

K

-

analytic

then

π

X

⁢

(

S

)

is

F

-analytic.

The proof of this follows easily from the definition of analytic sets.
First, there is a compact paved space

(

K

′

,

𝒦

′

)

and a set

T

∈

(

ℱ

×

𝒦

×

𝒦

′

)

σ

⁢

δ

such that

S

=

π

X

×

K

⁢

(

T

)

. Then,

π

X

⁢

(

S

)

=

π

X

⁢

(

π

X

×

K

⁢

(

T

)

)

=

π

X

⁢

(

T

)

.

However,

(

K

×

K

′

,

𝒦

×

𝒦

′

)

is a compact paved space (see

products of compact pavings are compact

(

http://planetmath.org/ProductsOfCompactPavingsAreCompact

)), which shows that

π

X

⁢

(

S

)

satisfies

the definition of

ℱ

-analytic sets.

Projections along Polish spaces

Theorem

1

above can be used to prove the following result for projections from the

product

of a

measurable space

and a

Polish space

. For

σ

-

algebras

(

http://planetmath.org/SigmaAlgebra

)

ℱ

and

ℬ

, we use the notation

ℱ

⊗

ℬ

for the product

σ

-algebra (

http://planetmath.org/ProductSigmaAlgebra

), in order to distinguish it from the product paving

ℱ

×

ℬ

.

Theorem 2

.

Let

(

X

,

F

)

be a measurable space and

Y

be a Polish space with Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

)

B

.

If

S

⊆

X

×

Y

is

F

⊗

B

-analytic, then its projection onto

X

is

F

-analytic.

An immediate

consequence

of this is the

measurable projection theorem

.

Although Theorem

2

applies to arbitrary Polish spaces, it is enough to just consider the case where

Y

is the space of

real numbers

ℝ

with the

standard topology

. Indeed, all Polish spaces are Borel

isomorphic

to either the real numbers or a

discrete subset

of the reals (see

Polish spaces up to Borel isomorphism

), so the general case follows from this.

If

Y

=

ℝ

, then the Borel

σ

-algebra is

generated by

the compact paving

𝒦

of closed and

bounded intervals

. The

collection

a

⁢

(

ℱ

×

𝒦

)

of analytic sets is

closed under

countable

unions and countable

intersections

so, by the

monotone class theorem

, includes the product

σ

-algebra

ℱ

⊗

ℬ

. Then, as the analytic sets define a

closure operator

,

a

⁢

(

ℱ

⊗

ℬ

)

⊆

a

⁢

(

a

⁢

(

ℱ

×

𝒦

)

)

=

a

⁢

(

ℱ

×

𝒦

)

.

Thus every

ℱ

⊗

ℬ

-analytic set is

ℱ

×

𝒦

-analytic, and the result follows from Theorem

1

.

Title

projections of analytic sets are analytic

Canonical name

ProjectionsOfAnalyticSetsAreAnalytic

Date of creation

2013-03-22 18:46:21

Last modified on

2013-03-22 18:46:21

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Related topic

MeasurableProjectionTheorem

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