In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as
$$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < \frac{k+1}{2^n}\right) + n1\left(n \leq Y\right)$$
Can you tell me why this is correct and why some other approximation would not work (or would it?)
Thank you very much.