A classical result of functional analysis states that the space $L^1(K, \mu)$, $\mu$ a nonatomic probability measure on a separable metric space $K$, is isometric to $L^1([0,1])$. The key ingredient for this is the existence of a bijection $\sigma\colon K\to[0,1]$ with the additional properties that $\sigma$ and $\sigma^{-1}$ are both Borel maps and that $\mu(B) = \lambda(\sigma^{-1}(B))$ for every Borel set $B\subset K$.
I am interested both in a reference to the standard proof of this result as well as a reference to the "original" proof in case it is known who gave the first proof of this result.