proof of extending a capacity to a Cartesian product

proof of extending a capacity to a Cartesian product

Let

(

X

,

ℱ

)

be a paved space such that

ℱ

is

closed under

finite unions and finite

intersections

, and

(

K

,

𝒦

)

be a

compact

paved space.
Define

𝒢

to be the

closure

under finite unions and finite intersections of the paving

ℱ

×

𝒦

on

X

×

K

.
For an

ℱ

-capacity

I

, define

I

~

:

𝒫

⁢

(

X

×

K

)

→

ℝ

,

I

~

⁢

(

S

)

=

I

⁢

(

π

X

⁢

(

S

)

)

,

where

π

X

is the

projection map

onto

X

. We show that

I

~

is a

𝒢

-capacity and that

π

X

⁢

(

S

)

∈

ℱ

δ

whenever

S

∈

𝒢

δ

.

Clearly, the property that

I

~

is an increasing

set function

follows from the fact that

I

satisfies

this property. Furthermore, if

S

n

⊆

X

×

K

is an

increasing sequence

of sets with

S

=

⋃

n

S

n

then

π

X

⁢

(

S

n

)

is an increasing sequence and

I

~

⁢

(

S

)

=

I

⁢

(

π

X

⁢

(

S

)

)

=

I

⁢

(

⋃

n

π

X

⁢

(

S

n

)

)

=

lim

n

→

∞

⁡

I

⁢

(

π

X

⁢

(

S

n

)

)

=

lim

n

→

∞

⁡

I

~

⁢

(

S

n

)

.

To prove that

I

~

is a

𝒢

-capacity, it only remains to show that if

S

n

is a

sequence

in

𝒢

decreasing to

S

⊆

X

×

K

then

I

~

⁢

(

S

n

)

→

I

~

⁢

(

S

)

.
Note that any

S

in

𝒢

can be written as

S

=

⋂

j

=

1

m

⋃

k

=

1

n

j

A

j

,

k

×

K

j

,

k

for sets

A

j

,

k

∈

ℱ

and

K

j

,

k

∈

𝒦

. The

projection

onto

X

is then

π

X

⁢

(

S

)

=

⋃

{

⋂

j

=

1

m

A

j

,

k

j

:

k

j

≤

n

j

,

⋂

j

=

1

m

K

j

,

k

j

≠

∅

}

which, as

ℱ

is closed under finite unions and finite intersections, must be in

ℱ

.
Furthermore, for any

x

∈

X

,

S

x

≡

{

y

∈

K

:

(

x

,

y

)

∈

S

}

=

⋂

j

=

1

m

⋃

{

K

j

,

k

:

k

≤

n

j

,

x

∈

A

j

,

k

}

.

This shows that

S

x

is in the closure

𝒦

*

of

𝒦

under finite unions and finite intersections. Furthermore, since compact pavings are

closed subsets

of a compact

topological space

(

http://planetmath.org/CompactPavingsAreClosedSubsetsOfACompactSpace

),

𝒦

*

is itself a compact paving.

Now let

S

n

be a

decreasing sequence

of sets in

𝒢

and set

S

=

⋂

n

S

n

. Then

π

X

⁢

(

S

)

⊆

π

X

⁢

(

S

n

)

for each

n

, giving

π

X

⁢

(

S

)

⊆

⋂

n

π

X

⁢

(

S

n

)

. To prove the reverse

inequality

, consider

x

∈

⋂

n

π

X

⁢

(

S

n

)

. Then,

(

S

n

)

x

is a

nonempty set

in

𝒦

*

for all

n

. By

compactness

,

S

x

=

⋂

n

(

S

n

)

x

must also be nonempty and therefore

x

∈

π

X

⁢

(

S

)

. This shows that

⋂

n

π

X

⁢

(

S

n

)

=

π

X

⁢

(

S

)

.

Furthermore, as we have shown that

π

X

⁢

(

S

n

)

∈

ℱ

and, as

I

is an

ℱ

-capacity,

I

~

⁢

(

S

n

)

=

I

⁢

(

π

X

⁢

(

S

n

)

)

→

I

⁢

(

π

X

⁢

(

S

)

)

=

I

~

⁢

(

S

)

.

So

I

~

is a

𝒢

-capacity.

We finally show that if

S

∈

𝒢

δ

then

π

X

⁢

(

S

)

∈

ℱ

δ

. By definition, there is a sequence

S

n

∈

𝒢

such that

S

=

⋂

n

S

n

. Setting

S

n

′

=

⋂

m

≤

n

S

m

then, since

𝒢

is closed under finite unions and finite intersections,

S

n

′

∈

𝒢

. Furthermore,

S

n

′

decreases to

S

so, as shown above,

π

X

⁢

(

S

n

′

)

∈

ℱ

and

π

X

⁢

(

S

)

=

⋂

n

π

X

⁢

(

S

n

′

)

∈

ℱ

δ

as required.

Title

proof of extending a capacity to a

Cartesian product

Canonical name

ProofOfExtendingACapacityToACartesianProduct

Date of creation

2013-03-22 18:47:41

Last modified on

2013-03-22 18:47:41

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Proof

Classification

msc 28A12

Classification

msc 28A05

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