# Sigma-Algebras as Information: Interactive Dice Lab

See a σ-algebra as what an observer can answer: two dice, blocks of outcomes they cannot tell apart, and events that are unions of blocks. Interactive lab.

## About this tool

A σ-algebra is the collection of events an observer can answer. Imagine throwing two dice while the observer sees only one number about them, such as the first die, the sum or the larger value. Outcomes with the same number look identical to the observer, so the 36 outcomes fall into blocks. An event is answerable exactly when it is a union of whole blocks. The first page lets you choose the observer and see which events they can settle. The second page shows information growing, one binary digit at a time.

## Frequently asked questions

### What is a σ-algebra?

It is a family of events that is closed under the operations you would expect to be able to do: if you can answer whether A happened, you can answer whether it did not; and if you can answer a countable list of questions, you can answer whether any of them happened. In plain words, it is the set of questions an observer can settle. With m blocks there are exactly 2ᵐ answerable events, one for each union of blocks.

### Why do blocks describe what an observer knows?

Because an observer who sees only a number cannot tell apart two outcomes that give the same number. Those outcomes form a block, and every question the observer can answer must get the same reply for all outcomes inside one block. So the answerable events are the unions of blocks. Seeing more detail splits blocks into smaller ones and enlarges the σ-algebra. Seeing nothing leaves one block, and only the empty event and the whole space can be answered.

### What should I try on the pages?

Switch between observers: nothing, the first die, the sum, the maximum, parities, everything. For each, pick an event such as both dice even or doubles and see whether it is answerable. Note which outcomes stay undecided: they sit in blocks that the event cuts in two. On the filtration page, reveal digits one at a time and watch the blocks halve while the conditional probability of an event moves towards 0 or 1.

### Is a σ-algebra the same as the information in a random variable?

Close, but it is the information, not the number. A random variable generates a σ-algebra: the events of the form {X is in B}. Two different variables, such as the sum and the sum plus one, can generate the same σ-algebra because they separate the same outcomes. The σ-algebra records which outcomes can be told apart, not what value they are labelled with. A filtration is such information growing in time.

## Learn the theory

- [σ-algebras and Borel sets](https://quantsandbox.dev/guide/probability/sigma-algebras-borel-sets): Which sets receive a size: σ-algebras as collections of answerable questions, partitions and atoms, generated σ-algebras, Borel sets and Vitali's set.
- [Independence and product measures](https://quantsandbox.dev/guide/probability/independence-product-measures): Independent events, σ-algebras and random variables, covariance and the variance of sums, product measures, Fubini's theorem and independent coin tosses.
- [Measure theory in option pricing](https://quantsandbox.dev/guide/probability/measure-theory-pricing): How the measure theory and probability of this track meet in option pricing: equivalent measures and densities, information, conditional expectation, limits.
- [Probability spaces, σ-algebras and random variables](https://quantsandbox.dev/guide/stochastic/probability-spaces): Sample space, σ-algebra, probability measure, measurable maps and the σ-algebra generated by a variable.
- [Filtrations, adapted processes and stopping times](https://quantsandbox.dev/guide/stochastic/filtrations): Information that grows over time, adapted and predictable processes, and when a random time is a stopping time.

Open the interactive tool: https://quantsandbox.dev/sigma-algebras