I use $\mu^*$ to denote the outer measure of a subset of $\mathbb{R}$.
Recently on a HW, I had a countable collection of measurable, pairwise disjoint sets {$E_k$}, and I wanted to show $\mu^*(A\cap\bigcup_kE_k)=\sum_k\mu^*(A\cap E_k)$, where $A$ is bounded.
In a previous HW, I proved that $\mu^*(A\cap (E_1\cup E_2))=\mu^*(A\cap E_1)+m^*(A\cap E_2)$.
So I used the latter equation as my base case, and WLOG, since my index set is countable, assumed that the index set for my collection of sets $E_k$ was the set $\mathbb{N}$.
My professor said I am not allowed to use induction here since induction only works for a finite number of objects.
Aren't all the dominoes supposed to fall? That's what the axiom of induction says:
- $P(i)$ Base Case ($i=2$ in my case)
- $\forall n. P(n)\Rightarrow P(n+1)$.
- Then $\forall n.P(n)$
Is induction only allowed for finite sets?