Generated Sigma Algebra Calculator

Enter Finite Universe and Generators

Use comma-separated elements for the universe. Enter one generator set per line. Every generator must be a subset of the universe.

Universe Elements

Duplicates are removed automatically.

Generator Sets

Each line creates one generator G_i.

What the Calculator Returns

Atom partition induced by the generators
Element membership signatures
Exact sigma algebra size for finite input
Full measurable-set list when atoms stay manageable
CSV and PDF export options

Reset

Universe	Generators	Atoms	Generated Sigma Algebra Size	Interpretation
{1, 2, 3, 4}	G1 = {1, 2} G2 = {2, 3}	{1}, {2}, {3}, {4}	16	The generators separate every element, so the full power set appears.
{a, b, c, d}	G1 = {a, b} G2 = {a, b}	{a, b}, {c, d}	4	Repeated generators do not refine atoms further.
{x, y, z}	G1 = {x}	{x}, {y, z}	4	Only unions of the two atoms are measurable.

Formula Used

The calculator works on a finite universe Ω and a generator family 𝒢 = {G₁, G₂, ..., G_m}.

x ~ y iff 1_Gᵢ(x) = 1_Gᵢ(y) for every generator Gᵢ

Elements with the same membership signature belong to the same atom. Every atom is a minimal nonempty block that cannot be split using the supplied generators.

A(signature) = ⋂ Hᵢ, where Hᵢ is either Gᵢ or Gᵢ^c

After the atoms A₁, A₂, ..., A_k are found, the generated sigma algebra is every possible union of those atoms.

σ(𝒢) = { ⋃ Aⱼ : choose any collection of atoms }

|σ(𝒢)| = 2^k, where k is the number of atoms

This finite construction gives the smallest sigma algebra containing all listed generators.

How to Use This Calculator

Enter the finite universe as comma-separated symbols, labels, or numbers.
Type one generator set per line using only elements already in the universe.
Press Generate Sigma Algebra to compute atoms and the resulting measurable family.
Review the atom table first, because every measurable set is built from those blocks.
Inspect the signature table to see why elements are grouped together.
Use the export buttons to save the result as CSV or PDF for notes, proofs, or assignments.

Frequently Asked Questions

1. What does this calculator actually compute?

It computes the smallest sigma algebra containing the generator sets on a finite universe. It also shows atoms, element signatures, and measurable sets when the atom count remains practical.

2. Why are atoms so important?

Atoms are the indivisible measurable blocks created by the generators. Every set in the generated sigma algebra is a union of atoms, so atoms completely determine the final structure.

3. Can I use letters instead of numbers?

Yes. The universe and generators accept text labels such as a, b, event1, or outcome_red. Duplicate labels are removed, and ordering is normalized for cleaner output.

4. What happens if a generator repeats?

Repeated generators do not change the final sigma algebra. They create the same membership information, so the atom partition stays unchanged unless a new generator refines it.

5. Why might the full measurable-set list be hidden?

The number of measurable sets doubles with every atom. Once the atom count becomes large, listing every set becomes bulky, so the calculator still reports the exact size instead.

6. Does the calculator test whether my generators already form a sigma algebra?

Yes. It checks for the empty set, the universe, complements, and finite unions within the supplied family. The summary then reports whether your generators already satisfy sigma-algebra closure.

7. Is this intended for infinite sample spaces?

No. This tool is designed for finite universes, which makes atom construction and enumeration explicit. It is best for teaching, finite probability models, and proof checking.

8. When do I get the full power set?

You get the full power set when the generators distinguish every element from every other element. In that case, each atom is a singleton, so every subset is measurable.