Almaz Khusnutdinov•Math enthusiast, love coding and mathematics
Published 6/9/2025
What is Sigma Algebra
Introduction
Lattices
Before diving into sigma-algebras, I’d like to start with the definition of a lattice.
A lattice is a partially ordered set (L,⪯) such that for all a,b∈L, the meet and join exist.
a∧b=inf{a,b}∈L,a∨b=sup{a,b}∈L
The meet a∧b satisfies:
a∧b⪯a,a∧b⪯b,and if x⪯a and x⪯b, then x⪯a∧b
That being said, a succeeds (a meet b) and if x precedes a and b, then it precedes the meet result.
The join a∨b satisfies:
a⪯a∨b,b⪯a∨b,and if a⪯y and b⪯y, then a∨b⪯y
Example - Hasse Diagrams
The Hasse diagram shows the divisibility relation on the set.
Each node represents an element of the set.
An edge from a node a to a node b means:
a∣band there is no c such that a∣c and c∣b with a=c=b
That said there is no element between those two, which is called immediate predecessor/successor.
The element 1 divides every other element:
1∣2,1∣3,1∣6
Elements 2 and 3 divide 6:
2∣6,3∣6
Thus, the diagram visually represents the partial order defined by divisibility on the set {1, 2, 3, 6}.
Boolean Algebra & Algebra of Sets
A Boolean algebra is a structure (B, ∧, ∨, ', 0, 1) where B is a set equipped with two binary operations meet ∧ and join ∨, a unary operation complement ', and distinguished elements 0 and 1.
Formally, B is a complemented distributive lattice, i.e.:
1. (B, ∧, ∨) is a lattice: for all a, b in B, the meet and join exist.
a∧b=inf{a,b}∈B,a∨b=sup{a,b}∈B
2. The lattice is distributive, meaning for all a, b, c in B,
a∧(b∨c)=(a∧b)∨(a∧c)
and
a∨(b∧c)=(a∨b)∧(a∨c)
3. There exist least and greatest elements 0, 1 in B such that for all a in B,
0⪯a⪯1
where ≤ is the partial order induced by the lattice operations:
a⪯b⟺a∧b=a(equivalently a∨b=b)
4. The lattice is complemented: for every element a in B, there exists a complement a' in B satisfying
a∧a′=0
and
a∨a′=1
Algebra of Sets
An algebra of sets on a non-empty set X is a non-empty collection A of subsets of X such that:
1. The whole set X is in A.
X∈A
2. A is closed under complementation relative to X:
∀A∈A⟹Ac=X∖A∈A
3. A is closed under finite unions:
∀A,B∈A⟹A∪B∈A
From these axioms, it follows that A is also closed under finite intersections:
∀A,B∈A⟹A∩B∈A
because
A∩B=(Ac∪Bc)c
Algebra of Sets as a specific case of Boolean Algebra
An algebra of sets A⊆P(X) with operations
a∧b=a∩b,a∨b=a∪b,a′=X∖a,
and distinguished elements
0=∅,1=X,
is a Boolean algebra since (A,∧,∨,′,0,1) is a complemented distributive lattice.
What is a Sigma Algebra?
The Definition
A sigma-algebra -
F⊆P(X)
is an algebra of sets closed under countable unions, i.e.,
{An}n=1∞⊆F⟹n=1⋃∞An∈F.
Since Fis an algebra, it is closed under complementation and finite unions, and hence also under countable intersections by De Morgan's laws.
Trivial Cases. Atoms. Partitions
The two extreme cases of a sigma-algebra on X are:
The trivialsigma-algebra: {∅,X}
The power set sigma-algebra: P(X).
An atom in a sigma-algebraF is a non-empty set A -
∀A∈F:B⊆A⟹B=∅ or B=A.
What is a Partition?
A partition of a set X is a collection of subsets satisfying
