resample_measure_preserving

The single-coordinate resample map is measure-preserving on a product cube. Let $\mu = \bigotimes_i \mu_i$ be a product of probability measures on $\prod_i \alpha_i$ (finite index set), and fix a coordinate $i$. The map $(\omega, \omega') \mapsto \omega^{(i \leftarrow \omega'_i)}$, which replaces the $i$-th coordinate of $\omega$ by $\omega'_i$ (leaving the other coordinates of $\omega$ unchanged), pushes the product measure $\mu \otimes \mu$ forward to $\mu$. Equivalently: resampling one coordinate of a draw from a product measure, using an independent copy, leaves the distribution unchanged. This is the measure-theoretic backbone of the resampling form of the Efron–Stein inequality: it makes $Z(\omega)$ and $Z(\omega^{(i \leftarrow \omega'_i)})$ identically distributed, so the resampled difference $Z - Z'_i$ is well-behaved.