13 Set Functions

Mr. Samopriya Basu

Objectives

I Measurable space

Objectives

I Measurable space
I Set function

Objectives

I Measurable space
I Set function
I Additive an

Objectives

I Measurable space
I Set function
I Additive and σ- additive set function
I Measures

Set function

I Let Ω be the reference set and A be a class of subsets of R Set function
I Let Ω be the reference set and A be a class of subsets of R
I (Ω, A): Measurable space

I A set function γ on (Ω, A) is an extended real valued function
defined on A i.e.

Set function

I Let Ω be the reference set and A be a class of subsets of R

I (Ω, A): Measurable space
I A set function γ on (Ω, A) is an extended real valued function  defined on A i.e.
γ : A → R? = [−∞, ∞]

Set function

I Let Ω be the reference set and A be a class of subsets of R

I (Ω, A): Measurable space

I A set function γ on (Ω, A) is an extended real valued function defined on A i.e.
γ : A → R? = [−∞, ∞]

I e.g. 1: Ω = {1, 2, 3, …..}, A = class of all subsets of Ω

γ(A) = number of elements in A

Let A = {1, 3, 5, 8, 9}, then γ(A) = 5
Let A = {1, 3, 5, 7, ….}, then γ(A) = ∞

Set function

I Let Ω be the reference set and A be a class of subsets of R
I (Ω, A): Measurable space
I A set function γ on (Ω, A) is an extended real valued function
defined on A i.e.
γ : A → R? = [−∞, ∞]
I e.g. 1: Ω = {1, 2, 3, …..}, A = class of all subsets of Ω
γ(A) = number of elements in A
Let A = {1, 3, 5, 8, 9}, then γ(A) = 5
Let A = {1, 3, 5, 7, ….}, then γ(A) = ∞
I e.g. 2: Ω = R, A = class of all intervals of type
(a, b]; −∞ ≤ a < b ≤ ∞
γ(a, b] = b − a
Then, γ(1, 3] = 2, γ(−∞, 3] = ∞

Additive and σ-additive set function
I Additive:

Additive and σ-additive set function
I Additive:

A set function γ defined on (Ω, A) is called additive if