Proof of Bonferroni Inequalities

Proof of Bonferroni Inequalities

Definitions and Notation

.

A

measure space

is a triple

(

X

,

Σ

,

μ

)

, where

X

is a set,

Σ

is a

σ

-algebra over

X

, and

μ

:

Σ

→

[

0

,

∞

]

is a

measure

, that is, a non-negative

function

that is

countably additive

. If

A

∈

Σ

, the

characteristic function

of

A

is the function

χ

A

:

X

→

R

defined by

χ

A

⁢

(

x

)

=

1

if

x

∈

A

,

χ

A

⁢

(

x

)

=

0

if

x

∉

A

. A

unimodal sequence

is a

sequence

of

real numbers

a

0

,

a

1

,

…

,

a

n

for which there is an index

k

such that

a

i

≤

a

i

+

1

for

i

<

k

and

a

i

≥

a

i

+

1

for

i

≥

k

.

The proof of the following easy lemma is left to the reader:

Lemma 1

.

If

a

0

≤

a

1

≤

…

≤

a

k

≥

a

k

+

1

≥

a

k

+

2

≥

…

≥

a

n

is a unimodal sequence of non-negative real numbers with

∑

i

=

0

n

(

-

1

)

i

⁢

a

i

=

0

, then

∑

i

=

0

j

(

-

1

)

i

⁢

a

i

≥

0

for even

j

and

≤

0

for odd

j

.

Since the

binomial

sequence

(

(

a

i

)

)

0

≤

i

≤

n

with

integer

a

>

0

and integer

n

≥

a

satisfies

the

hypothesis

of Lemma

1

, we have:

Corollary 1

.

If

a

is a

positive integer

,

∑

i

=

0

j

(

-

1

)

i

⁢

(

a

i

)

≥

0

for even

j

and

≤

0

for odd

j

.

Lemma 2

.

Let

(

A

i

)

1

≤

i

≤

n

be a sequence of sets and let

X

=

⋃

1

≤

i

≤

n

A

i

. For

x

∈

X

, let

I

⁢

(

x

)

be the set of indices

j

such that

x

∈

A

j

. If

1

≤

k

≤

n

,

∑

1

≤

i

1

<

i

2

<

…

<

i

k

≤

n

χ

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

⁢

(

x

)

=

(

|

I

⁢

(

x

)

|

k

)

for all

x

∈

X

.

Proof.

χ

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

⁢

(

x

)

=

1

if

{

i

1

,

i

2

,

…

,

i

k

}

⊆

I

⁢

(

x

)

, and

=

0

otherwise. Therefore the sum equals the number of

k

-subsets of

I

⁢

(

x

)

, which is

(

|

I

⁢

(

x

)

|

k

)

.
∎

Theorem 1

.

Let

(

X

,

Σ

,

μ

)

be a measure space. If

(

A

i

)

1

≤

i

≤

n

is a

finite sequence

of

measurable sets

all having finite measure, and

S

j

=

μ

⁢

(

A

1

∪

A

2

∪

…

∪

A

n

)

+

∑

k

=

1

j

(

-

1

)

k

⁢

∑

1

≤

i

1

<

i

2

<

…

<

i

k

≤

n

μ

⁢

(

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

)

then

S

j

≥

0

for even

j

, and

≤

0

for odd

j

. Moreover,

S

n

=

0

(

Principle of Inclusion-Exclusion

).

Proof.

Let

Y

=

⋃

1

≤

i

≤

n

A

i

.

S

j

=

∫

Y

𝑑

μ

+

∑

k

=

1

j

(

-

1

)

k

⁢

∑

1

≤

i

1

<

i

2

<

…

<

i

k

≤

n

∫

Y

χ

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

⁢

𝑑

μ

=

∫

Y

𝑑

μ

+

∑

k

=

1

j

(

-

1

)

k

⁢

∫

Y

(

∑

1

≤

i

1

<

i

2

<

…

<

i

k

≤

n

χ

A

i

1

∩

A

i

2

∩

…

∩

A

i

k

)

⁢

𝑑

μ

By Lemma

2

,

S

j

=

∫

Y

𝑑

μ

+

∑

k

=

1

j

(

-

1

)

k

⁢

∫

Y

(

|

I

⁢

(

x

)

|

k

)

⁢

𝑑

μ

=

∑

k

=

0

j

(

-

1

)

k

⁢

∫

Y

(

|

I

⁢

(

x

)

|

k

)

⁢

𝑑

μ

=

∫

Y

∑

k

=

0

j

(

-

1

)

k

⁢

(

|

I

⁢

(

x

)

|

k

)

⁢

d

⁢

μ

Since

|

I

⁢

(

x

)

|

>

0

for

x

∈

Y

, it follows from Corollary

1

that, in the last

integral

, the integrand is

≥

0

for even

j

and

≤

0

for odd

j

. Therefore the same is true for the integral itself. In

addition

, the integrand is identically 0 for

j

=

n

, hence

S

n

=

0

.
∎

This proof shows that at the heart of Bonferroni’s

inequalities

lie

similar

inequalities governing the

binomial coefficients

.

Title

Proof of Bonferroni Inequalities

Canonical name

ProofOfBonferroniInequalities

Date of creation

2013-03-22 19:12:44

Last modified on

2013-03-22 19:12:44

Owner

csguy (26054)

Last modified by

csguy (26054)

Numerical id

6

Author

csguy (26054)

Entry type

Proof

Classification

msc 60A99

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