Lecture # 17
Algebra of Sets
In this lecture, we shall discuss algebra of sets and their properties.
Ring of sets
Let 𝑋 be a non empty set and ℛ be a collection of subsets of 𝑋. Then ℛ is said
to be ring of sets if
1. For all 𝐴, 𝐵 ∈ ℛ ⇒ 𝐴 ∪ 𝐵 ∈ ℛ
2. For all 𝐴, 𝐵 ∈ ℛ ⇒ 𝐴/𝐵 ∈ ℛ .
Examples:
1. Let 𝑋 = {1, 2, 3, 4} and ℛ 1 = {∅, 𝑋 , {1}, {2,3,4}}, ℛ 2 = {∅, {1},
{2}, {2,3,4}} are ring of sets on 𝑋.
2. For any non empty set 𝑋, ℛ = 𝑃(𝑋) is a ring of sets.
Remark:
Union of two ring of sets need not be a ring of sets.
Sigma Ring
A ring of sets ℛ on a set 𝑋 is said to be 𝜎 −ring if for every sequence {𝐴𝑛 } of sets
in ℛ implies ∪ 𝐴𝑛 ∈ ℛ.
e.g. For any non empty set 𝑋, ℛ = 𝑃(𝑋) is a sigma ring of sets.
Remark:
Every sigma ring is a ring but converse is not true.e.g, let 𝑋 = 𝑁 and
ℛ = {𝐴: 𝐴 ⊆ 𝑁 𝑎𝑛𝑑 𝐴 𝑖𝑠 𝑓𝑖𝑛𝑖𝑡𝑒}. Then ℛ is a ring but not a 𝜎 −ring.
Algebra on sets
Let 𝑋 be a non empty set and 𝒜 be a collection of subsets of 𝑋. Then 𝒜 is said
to be algebra on 𝑋 if
3. For all 𝐴, 𝐵 ∈ 𝒜 ⇒ 𝐴 ∪ 𝐵 ∈ 𝒜
4. For all 𝐴 ∈ 𝒜 ⇒ 𝐴′ ∈ 𝒜.
Example 1:
Let 𝑋 = {1, 2, 3, 4} and 𝒜1 = {∅, 𝑋 {1}, {2,3,4}}, 𝒜2 = {∅, 𝑋, {1, 2}, {3,4}}
and 𝒜3 = {∅, 𝑋, {1}, {3,4}}. Then 𝒜1 and 𝒜2 are algebras on 𝑋 but 𝒜3
is not an algebra on 𝑋.
Remark:
(i)
(ii)
(iii)
If 𝒜 is an algebra on 𝑋 and 𝐴, 𝐵 ∈ 𝒜 then 𝐴 ∩ 𝐵 ∈ 𝒜.
For any 𝑋 ≠ ∅, {∅, 𝑋} and 𝑃(𝑋) are algebra on 𝑋.
Let 𝑋 ≠ ∅ and 𝐴 ⊆ 𝑋. Then {∅, 𝑋, 𝐴, 𝐴′ } is an algebra on 𝑋.
Sigma algebra
An algebra 𝒜 on a set 𝑋 is said to be 𝜎 −algebra if for every sequence {𝐴𝑛 } of sets
in 𝒜 implies ∪ 𝐴𝑛 𝒜.
Measurable space
A measurable space is an ordered pair (𝑋, 𝒜), where 𝑋 is a non empty set and 𝒜 is
a 𝜎 −algebra.
Remark
1. Every 𝜎 −algebra is a 𝜎 −ring but converse is not true
2. Every 𝜎 −algebra is algebra but converse is not true.
Example: Show that if 𝑋 is a non empty set then 𝒜 = {𝐴 ⊆ 𝑋: 𝐴 𝑜𝑟 𝐴′ 𝑖𝑠 𝑓𝑖𝑛𝑖𝑡𝑒}
is an algebra on 𝑋.
Solution:
Let 𝐴, 𝐵 ∈ 𝒜 ⇒ 𝐴 𝑜𝑟 𝐴′ and 𝐵 𝑜𝑟 𝐵′ is finite.
Case 1: If 𝐴 𝑎𝑛𝑑 𝐵 are finite⇒ 𝐴 ∪ 𝐵 is finite ⇒ 𝐴 ∪ 𝐵 ∈ 𝒜.
Case 2: If 𝐴′ and 𝐵 are finite ⇒ (𝐴 ∪ 𝐵)′ ⇒ 𝐴′ ∩ 𝐵′ is finite because𝐴′ is finite⇒
𝐴 ∪ 𝐵 ∈ 𝒜.
Similarly in all other cases 𝐴 ∪ 𝐵 ∈ 𝒜.
Next, if 𝐴 ∈ 𝒜 ⇒ 𝐴 𝑜𝑟 𝐴′ is finite ⇒ 𝐴′ ∈ 𝒜. So 𝒜 is an algebra.
Exercise
Question #1:
Show that if 𝑋 = 𝑁 then 𝒜 = {𝐴 ⊆ 𝑋: 𝐴 𝑜𝑟 𝐴′ 𝑖𝑠 𝑓𝑖𝑛𝑖𝑡𝑒} is an algebra on 𝑁 but
not a sigma algebra.
Question #2:
Show that if 𝑋 is an uncountable set then 𝒜 = {𝐴 ⊆ 𝑋: 𝐴 𝑜𝑟 𝐴′ 𝑖𝑠 𝑐𝑜𝑢𝑛𝑡𝑎𝑏𝑙𝑒} is
a sigma algebra.