analytic set

analytic set

(Note: this entry concerns

analytic sets

as used in

measure theory

. For the

definition

in analytic spaces see analytic set (

http://planetmath.org/AnalyticSet

)).

For a

continuous map

of

topological spaces

it is known that the

preimages

of

open sets

are open, preimages of closed (

http://planetmath.org/ClosedSet

) sets are closed and preimages of

Borel sets

are themselves

Borel measurable

. The situation is more difficult for

direct images

(

http://planetmath.org/DirectImage

). That is if

f

:

X

→

Y

is

continuous

then it does not follow that

S

⊆

X

being open/closed/

measurable

implies the same

property

for

f

⁢

(

S

)

.
In fact,

f

⁢

(

X

)

=

Image

⁡

(

f

)

need not even be measurable.
One of the few things that can be said, however, is that

f

⁢

(

S

)

is

compact

whenever

S

is compact.

Analytic sets

are defined in order to be stable under direct images, and their theory relies on the stability of compact sets. This is a fruitful

concept

because, as it turns out, all

measurable sets

are

analytic

and all analytic sets are

universally measurable

.

A subset

S

of a

Polish space

X

is said to be analytic (or, a

Suslin set

) if it is the image of a continuous map

f

:

Z

→

X

from another Polish space

Z

— see Cohn. It is then clear that

g

⁢

(

S

)

will again be analytic for any continuous map

g

:

X

→

Y

between Polish spaces. Indeed,

g

⁢

(

S

)

will be the image of

g

∘

f

.

Here, we instead give the following definition which applies to arbitrary

paved spaces

and, in particular, to all

measurable spaces

. Furthermore, for Polish spaces, it can be shown to be

equivalent

to the definition just mentioned above.

Recall that for a paved space

(

X

,

ℱ

)

,

ℱ

σ

⁢

δ

represents the

collection

of

countable

intersections

of countable unions of elements of

ℱ

, and that a paving is compact if every subcollection satisfying the

finite intersection property

has nonempty intersection.

Definition

.

Let

(

X

,

F

)

be a paved space. Then a set

A

⊆

X

is said to be

F

-

analytic

, or

analytic with respect to

F

, if there exists a compact paved space

(

K

,

K

)

and an

S

∈

(

F

×

K

)

σ

⁢

δ

such that

A

=

π

⁢

(

S

)

.

Here,

F

×

K

is the product paving and

π

:

X

×

K

→

X

is the

projection map

π

⁢

(

x

,

y

)

=

x

.

Writing this out explicitly, there are doubly indexed

sequences

of sets

A

m

,

n

∈

ℱ

and

K

m

,

n

∈

𝒦

such that

S

=

⋂

n

=

1

∞

⋃

m

=

1

∞

A

m

,

n

×

K

m

,

n

.

and,

A

=

{

x

∈

X

:

(

x

,

y

)

∈

S

⁢

for some

⁢

y

∈

K

}

.

Although this allows for

𝒦

to be any compact paving it can be shown that it makes no

difference

if it is just taken to be the collection of compact subsets of, for example, the

real numbers

.

For a measurable space

(

X

,

ℱ

)

, a subset of

X

is simply said to be analytic if it is

ℱ

-analytic, and a subset of a topological space is said to be analytic if it is analytic with respect to the Borel

σ

-

algebra

(

http://planetmath.org/BorelSigmaAlgebra

).

References

1

K. Bichteler,

Stochastic integration

with jumps

. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.

2

Donald L. Cohn,

Measure theory

. Birkhäuser, 1980.

3

Claude Dellacherie, Paul-André Meyer,

Probabilities

and

potential

. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., 1978.

4

Sheng-we He, Jia-gang Wang, Jia-an Yan,

Semimartingale

theory and stochastic calculus.

Kexue Chubanshe (Science Press), CRC Press, 1992.

5

M.M. Rao,

Measure theory and integration

. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 265. Marcel Dekker Inc., 2004.

Title

analytic set

Canonical

name

AnalyticSet1

Date of creation

2013-03-22 18:44:52

Last modified on

2013-03-22 18:44:52

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

9

Author

gel (22282)

Entry type

Definition

Classification

msc 28A05

Synonym

Suslin set

Related topic

PavedSpace

Related topic

BorelSigmaAlgebra

Related topic

UniversallyMeasurable

Defines

Suslin set

Defines

analytic with respect to

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