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202405240847
Status: #idea
Tags: Probability Theory, Measure Theory
State: #nascient

The Saviour of Algebra : The σ−Algebra

Well, all a σ−algebra is, is some algebra which is closed under countably infinite unions. In other words, the following is true:

If A1,A2,… are all in the algebra Ξ under consideration, then:⋃i=1∞Ai=⋃i∈NAi∈Ξ

This addition is all we lacked to start developing Probability Theory.
Is this true for the intersections as well?

Please be aware that the σ−algebra can contain a finite number of elements, or an uncountable infinite number of elements, as long as it's close under infinite unions we are gucci.

Also σ−algebra IS an algebra, after all the condition is stronger. You can convince yourself that the converse is not true.

Subsets which belong to a σ−algebra, here denoted Ξ are called Ξ−measurable sets.

Countable vs Uncountable

Countable Sample Space

If our sample space Ω is countable (finite or infinite), it is possible to define a σ−algebra which simply consists of ALL possible subsets of Ω (denoted 2Ω), and therefore we can assign a probability to each outcome in the sample space. In such cases we are operating in what's called a discrete probability space.

This is important, because in such spaces even though the σ−algebra might be uncountably infinite (see Diagonalization Argument), it is straightforward to define a probability function such that ∑x∈Ωpx=1 and then assign to each event the probability of the sum of its elements. This is what we call a Discrete Probability Measure

Uncountable Sample Space

If the sample space is uncountably infinite, no such trick exist. That's when σ−algebras really come into their own and Borel Sets really come into their own.