measurable section theorem

measurable section theorem

Consider a

Cartesian product

X

×

Y

with

projection map

π

X

:

X

×

Y

→

X

. Then, a

cross section

of a subset

S

⊆

X

×

Y

is a map

τ

:

π

X

⁢

(

S

)

→

Y

satisfying

(

x

,

τ

⁢

(

x

)

)

∈

S

.
The following

theorem

states that such cross sections can be chosen in a measurable (

http://planetmath.org/MeasurableFunctions

) way. 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

.

Let

(

X

,

F

)

be a

universally complete

measurable space

(

http://planetmath.org/MeasurableSpace

),

Y

be a

Polish space

with Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

)

B

, and

π

X

:

X

×

Y

→

X

be the projection map.

For any

F

⊗

B

-

analytic set

S

, then

π

X

⁢

(

S

)

∈

F

and there is a measurable map

τ

:

π

X

⁢

(

S

)

→

Y

such that

(

x

,

τ

⁢

(

x

)

)

∈

S

for all

x

∈

π

X

⁢

(

S

)

.

The

existence

of a map satisfying

(

x

,

τ

⁢

(

x

)

)

∈

S

is no surprise, and indeed must exist by a simple application of the

axiom of choice

. The main force of the theorem is that

τ

may be taken to be measurable, which is not something that could be achieved by such basic use of choice.
In fact, the result fails to be true if either the

universal

completeness

assumption

for

(

X

,

ℱ

)

or the requirement that

Y

be a Polish space is dropped.

This result has important applications to continuous-time stochastic processes, where the space

Y

is taken to be the time

index set

, usually the set

ℝ

+

of nonnegative

real numbers

.

Consider

stochastic processes

(

U

t

)

t

∈

ℝ

+

and

(

V

t

)

t

∈

ℝ

+

defined on a

probability space

(

Ω

,

ℱ

,

ℙ

)

. Suppose that they were not

equivalent

so that, with positive

probability

, there is a time

t

at which they differ. Then the projection

π

Ω

⁢

(

S

)

of the set of times

S

=

{

(

t

,

ω

)

:

U

t

⁢

(

ω

)

≠

V

t

⁢

(

ω

)

}

has positive probability. Consequently, the

measurable section theorem

guarantees the existence of a random time

τ

for which

U

τ

≠

V

τ

with positive probability.

Corollary

.

Let

(

U

t

)

t

∈

R

+

and

(

V

t

)

t

∈

R

+

be jointly measurable stochastic processes defined on the probability space

(

Ω

,

F

,

P

)

. If

ℙ

(

U

τ

=

V

τ

)

=

1

for all random times

τ

:

Ω

→

R

+

, then

U

and

V

are equivalent processes.

So, continuous-time stochastic processes are fully determined up to equivalence by their values, neglecting zero probability sets, at random times. Note that the condition that

(

Ω

,

ℱ

)

be universally complete can be dropped from the statement, due to the fact that any

random variable

τ

on the

completion

of a probability space can be replaced by a random variable

τ

′

satisfying

ℙ

(

τ

=

τ

′

)

=

1

.

More useful statements which apply to

adapted processes

, and allow

τ

to be either a

stopping time

or a

predictable stopping time

, are given by the

optional

and

predictable

section

theorems.

Title

measurable section theorem

Canonical name

MeasurableSectionTheorem

Date of creation

2013-03-22 18:48:21

Last modified on

2013-03-22 18:48:21

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A05

Synonym

measurable cross section theorem

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