continuous almost everywhere versus equal to a continuous function almost everywhere

continuous almost everywhere versus equal to a continuous function almost everywhere

The

concept

of almost everywhere can be somewhat tricky to people who are not familiar with it. Let

m

denote

Lebesgue measure

. Consider the following two statements about a

function

f

:

ℝ

→

ℝ

:

•

f

is

continuous

(

http://planetmath.org/Continuous

) almost everywhere with respect to

m

•

f

is equal to a

continuous function

almost everywhere with respect to

m

Although these two statements seem alike, they have quite different meanings. In fact, neither one of these statements implies the other.

Consider the function

χ

[

0

,

∞

)

⁢

(

x

)

=

{

1

if

⁢

x

≥

0

0

if

⁢

x

<

0

.

This function is not continuous at

0

, but it is continuous at all other

x

∈

ℝ

. Note that

m

⁢

(

{

0

}

)

=

0

. Thus,

χ

[

0

,

∞

)

is continuous almost everywhere.

Suppose

χ

[

0

,

∞

)

is equal to a continuous function almost everywhere. Let

A

⊂

ℝ

be Lebesgue measurable (

http://planetmath.org/LebesgueMeasure

) with

m

⁢

(

A

)

=

0

and

g

:

ℝ

→

ℝ

such that

χ

[

0

,

∞

)

⁢

(

x

)

=

g

⁢

(

x

)

for all

x

∈

ℝ

∖

A

. Since

χ

[

0

,

∞

)

⁢

(

x

)

=

0

for all

x

<

0

and

m

⁢

(

A

∩

(

-

∞

,

0

)

)

=

0

, there exists

a

<

0

such that

g

⁢

(

a

)

=

0

. Similarly, there exists

b

≥

0

such that

g

⁢

(

b

)

=

1

. Since

g

is continuous, by the

intermediate value theorem

, there exists

c

∈

(

a

,

b

)

with

g

⁢

(

c

)

=

1

2

. Let

U

=

(

0

,

1

)

. Since

g

is continuous,

g

-

1

⁢

(

U

)

is open. Recall that

c

∈

g

-

1

⁢

(

U

)

. Thus,

g

-

1

⁢

(

U

)

≠

∅

. Since

g

-

1

⁢

(

U

)

is a nonempty open set,

m

⁢

(

g

-

1

⁢

(

U

)

)

>

0

. On the other hand,

g

-

1

⁢

(

U

)

⊆

A

, yielding that

0

<

m

⁢

(

g

-

1

⁢

(

U

)

)

≤

m

⁢

(

A

)

=

0

, a

contradiction

.

Now consider the function

χ

ℚ

⁢

(

x

)

=

{

1

if

⁢

x

∈

ℚ

0

if

⁢

x

∉

ℚ

.

Note that

m

⁢

(

ℚ

)

=

0

. Thus,

χ

ℚ

=

0

almost everywhere. Since

0

is continuous,

χ

ℚ

is equal to a continuous function almost everywhere. On the other hand,

χ

ℚ

is not continuous almost everywhere. Actually,

χ

ℚ

is not continuous at any

x

∈

ℝ

. Recall that

ℚ

and

ℝ

∖

ℚ

are both dense in (

http://planetmath.org/Dense

)

ℝ

. Therefore, for every

x

∈

ℝ

and for every

δ

>

0

, there exist

x

1

∈

(

x

-

δ

,

x

+

δ

)

∩

ℚ

and

x

2

∈

(

x

-

δ

,

x

+

δ

)

∩

(

ℝ

∖

ℚ

)

. Since

χ

ℚ

⁢

(

x

1

)

=

1

and

χ

ℚ

⁢

(

x

2

)

=

0

, it follows that

χ

ℚ

is not continuous at

x

. (Choose any

ε

∈

(

0

,

1

)

.)

Title

continuous almost everywhere versus equal to a continuous function almost everywhere

Canonical name

ContinuousAlmostEverywhereVersusEqualToAContinuousFunctionAlmostEverywhere

Date of creation

2013-03-22 15:58:47

Last modified on

2013-03-22 15:58:47

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 28A12

Classification

msc 60A10

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