Probability Tidbits 1 - Banach Tarski

Not all sets are measurable. Within [0,1] there exists unmeasurable sets. We’ll construct one here. First define an equivalence class x∼y⟺x−y∈Q. The set [0,1] contains many such equivalence classes(which by definition are disjoint from each other). By axiom of choice, we can choose a representative from each equivalence class. Let A be a set containing the representatives. We construct a disjoint cover:

[0,1]⊂∪r∈Q[−1,1]{a+r:a∈A}:=U

Where Q[−1,1]:={x:x∈Q,−1≤x≤1}. Denote each Xr:={a+r:a∈A}. U covers [0,1] because ∀x∈[0,1],x∼a for some a∈A by definition, and so x−a∈Q[−1,1]. For r≠s, Xr∩Xs=∅ because suppose they’re not disjoint, then:

x∈Xr∩Xs⟹∃ar,as∈A s.t. ar+r=as+s=x⟹ar−as=s−r∈Q⟹ar∼as⟹ar=as⟹r=s

Which is a contradiction. Also, U⊂[−1,2] because

r≥−1,a≥0⟹a+r≥−1∀a∈A,r∈Q[−1,1]

r≤1,a≤1⟹a+r≤2∀a∈A,r∈Q[−1,1]

Because lebesgue measures are translation invariant:

μ(Xr)=μ(A)∀r∈Q[−1,1]⟹μ([0,1])≤μ(∪r∈Q[−1,1]Xr)≤μ([−1,2])1≤Σr∈Q[−1,1]μ(A)≤3 which is impossible.

Knowing that some sets aren’t measurable is important - it means we can’t assign a probability to specific sets of outcomes. We need a more rigorous definition for probability theory, and that’s where measure theory comes in.