STATEMENT: Let $α$ be the non-decreasing function on $\mathbb R$ defined by $α(t) = 0$ if $t ≤ 0$ and $α(t) = 1$ if $t > 0$. Let $μ_α([a,b))=\alpha(b)-\alpha(a)$ , with $μ^*_α$ the corresponding outer measure. Determine which sets are in $M(μ^∗_α)$, that is, measurable for this outer measure.
QUESTION: I have found sets which are in $M(\mu^*_\alpha)$. My question is how do I show that what I have found is in fact all of the measurable sets? Some of the sets are for example $(a,b),[a,b),(a,b],[a,b]$ where $a\leq 0$.