Fubini’s theorem for the Lebesgue integral

Fubini’s theorem for the Lebesgue integral

This is the version of the Fubini’s Theorem for the

Lebesgue integral

. For
the

Riemann integral

, see the standard

calculus

version (

http://planetmath.org/FubinisTheorem

).

In the following suppose we will by convention define

∫

X

f

⁢

𝑑

μ

:=

0

in case

f

is not

integrable

. This simplifies notation and does not affect the results since it will turn out that such cases happen on a set of

measure

0.

Also if we have a function

f

:

X

×

Y

→

𝔽

then define

f

x

⁢

(

y

)

:=

f

⁢

(

x

,

y

)

and

f

y

⁢

(

x

)

:=

f

⁢

(

x

,

y

)

.

Theorem

(Fubini).

Suppose

(

X

,

M

,

μ

)

and

(

Y

,

N

,

ν

)

are

σ

-finite (

http://planetmath.org/SigmaFinite

)

measure spaces

. If

f

∈

L

1

⁢

(

X

×

Y

)

then

f

x

∈

L

1

⁢

(

ν

)

for

μ

-almost every

x

and

f

y

∈

L

1

⁢

(

μ

)

for

ν

-almost every

y

. Further
the functions

x

↦

∫

Y

f

x

⁢

𝑑

ν

and

y

↦

∫

X

f

y

⁢

𝑑

ν

are
in

L

1

⁢

(

μ

)

and

L

1

⁢

(

ν

)

respectively and

∫

X

×

Y

f

⁢

d

⁢

(

μ

×

ν

)

=

∫

X

[

∫

Y

f

⁢

(

x

,

y

)

⁢

𝑑

ν

⁢

(

y

)

]

⁢

𝑑

μ

⁢

(

x

)

=

∫

Y

[

∫

X

f

⁢

(

x

,

y

)

⁢

𝑑

μ

⁢

(

x

)

]

⁢

𝑑

ν

⁢

(

y

)

.

You can now see the reason for defining the integral even where

f

x

and

f

y

are not integrable since the functions

x

↦

∫

Y

f

x

⁢

𝑑

ν

and

y

↦

∫

X

f

y

⁢

𝑑

ν

are normally only almost everywhere defined, and we’d like to define them everywhere. Since we have changed the definition only on a set of

measure
zero

, this does not change the final result and we can interchange the
integrals freely without having to worry about where the functions are
actually defined.

Note the of this theorem and Tonelli’s theorem for non-negative functions. Here you actually need to check some integrability before switching
the integral . A application of Tonelli’s theorem actually shows that you can prove any one of these equations to show that

f

∈

L

1

⁢

(

X

×

Y

)

∫

X

×

Y

|

f

|

⁢

d

⁢

(

μ

×

ν

)

<

∞

,

∫

X

[

∫

Y

|

f

⁢

(

x

,

y

)

|

⁢

𝑑

ν

⁢

(

y

)

]

⁢

𝑑

μ

⁢

(

x

)

<

∞

,

or

∫

Y

[

∫

X

|

f

⁢

(

x

,

y

)

|

⁢

𝑑

μ

⁢

(

x

)

]

⁢

𝑑

ν

⁢

(

y

)

<

∞

.

If we take the

counting measure

on

ℕ

, then one can the

Fubini theorem

for sums.

Theorem

(Fubini for sums).

Suppose that

f

i

⁢

j

is absolutely summable, that is

∑

i

,

j

∈

N

|

f

i

⁢

j

|

<

∞

, then

∑

i

,

j

∈

ℕ

f

i

⁢

j

=

∑

i

=

1

∞

∑

j

=

1

∞

f

i

⁢

j

=

∑

j

=

1

∞

∑

i

=

1

∞

f

i

⁢

j

.

In the above theorem we have used

ℕ

as our set for simplicity and familiarity of notation. Any

summable function

f

i

⁢

j

will have only a countable number of non-zero elements and thus the theorem for arbitrary
sets just reduces to the above case.

References

1

Gerald B. Folland.

. John Wiley & Sons, Inc., New York, New York, 1999

Title

Fubini’s theorem for the Lebesgue integral

Canonical name

FubinisTheoremForTheLebesgueIntegral

Date of creation

2013-03-22 14:16:09

Last modified on

2013-03-22 14:16:09

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Theorem

Classification

msc 28A35

Synonym

Fubini’s theorem

Related topic

FubinisTheorem

Related topic

TonellisTheorem

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